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2017 2



A study on consecutive edge-magic labelings and safe sets through graph operations
( edge-magic safe set)


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A study on consecutive edge-magic labelings and safe sets through graph
operations
Doctor of Philosophy
Seoul National University
Department of Mathematics Education
Abstract
In this thesis, we utilize graph operations to study consecutive edge-magic labelings and
safe sets of graphs.
An edge-magic total labeling f of a graph G is a bijective function from V (G) ∪E(G)
to {1, . . . , |V (G)| + |E(G)|} such that for each uv ∈ E(G), f(u) + f(v) + f(uv) = c
for some positive integer c. If f(E(G)) is a set of consecutive integer, then f is called a
consecutive edge-magic labeling and G is said to be consecutive edge-magic. Especially,
if f(E(G)) = {|V (G)| + 1, . . . , |V (G)| + |E(G)|}, then f is called a super edge-magic
labeling and G is said to be super-edge magic. The notion of b-edge consecutive magic
labeling is derived from the notion of super edge-magic labeling, that is, if a consecutive
edge-magic labeling f of G satisfies f(E(G)) = {b+ 1, . . . , b+ |E(G)|}, then f is called
a b-edge consecutive magic labeling and G is said to be b-edge consecutive magic.
Study on consecutive edge-magic labelings is one of the most popular topic on the
study of graph labeling. Especially, characterizing consecutive edge-magic graphs is one
of the most popular topic in the study of consecutive edge-magic labeling. Though there
has been much study on the topic, determining whether or not it yields a consecutive edge-
magic labeling remains open even for graphs with rather simple structures. In this thesis,
we first study properties of consecutive edge-magic graphs and construct new consecutive
edge-magic graph from existing consecutive edge-magic graphs through graph operation.
We find out many special properties that a graph G which admits b-edge consecutive magic
labeling for 0 < b < |V (G)| has. In particular, we show that a graph G admitting b-edge
consecutive magic labeling for 0 < b < |V (G)| is a graceful tree having super edge-magic
labeling. Then we provide several ways of obtaining new consecutive edge-magic graphs
from consecutive edge-magic graphs through graph operation. Especially, we show that
there are infinitely many super edge-magic trees containing arbitrary number of copies of
a given super edge-magic tree. Those methods of obtaining new consecutive edge-magic
graphs we developed are differentiated from other studies and expected to contribute to
i
ii
settling many open questions related to consecutive edge-magic labeling.
We also study the safe number and the connected safe number of Cartesian product of
two complete graphs. For a connected graph G, a vertex subset S of V (G) is said to be a safe
set if for every component C of the subgraph of G induced by S, |C| ≥ |D| holds for every
component D of G− S such that there exists an edge between C and D, and, in particular,
if the subgraph induced by S is connected, then S is called a connected safe set. For a
connected graph G, the safe number and the connected safe number of G are the minimum
among sizes of safe sets and the minimum among sizes of connected safe sets, respectively,
of G. We show that the safe number and the connected safe number of Cartesian product
of two complete graphs are equal and present a polynomial-time algorithm to compute the
connected safe number of Cartesian product of two complete graphs. We believe that our
idea used in the analysis of the structure of the graphs is useful enough to be applied to other
graphs.
labeling, adjacent matrix, safe set, connected safe set
Student Number: 2011-23647
1.2 Magic labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Super edge-magic labelings and Enomoto’s conjecture . . . 14
1.2.3 b-edge consecutive magic labelings . . . . . . . . . . . . . 18
1.3 Safe sets of graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 A preview of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 On consecutive edge-magic labeling of a tree 25
2.1 Consecutive edge magic total labelings of connected bipartite graphs 25
2.1.1 Properties of edge-magic total labelings . . . . . . . . . . . 25
2.1.2 Consecutive edge magic labelings for trees . . . . . . . . . 35
2.2 Super edge-magic labeling graphs . . . . . . . . . . . . . . . . . . 44
2.2.1 Obtaining a new super edge-magic graph through graph op-
erations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3 b-edge consecutive magic labeling . . . . . . . . . . . . . . . . . . 62
iii
(X, Y ) and its labeling . . . . . . . . . . . . . . . . . . . . 69
2.4.2 Number of super edge-magic graphs and its labelings . . . . 80
3 A study on safe sets of a graph 90
3.1 The safe number and the connected safe number of Cartesian prod-
uct of two complete graphs . . . . . . . . . . . . . . . . . . . . . . 91
3.2 Simplified formulae for the safe numbers of KmKn for m ∈
{3, 4}, n ≥ m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Abstract (in Korean) 121
1.1 Basic definitions and notations of graph theory
In this section, we introduce basic definitions and notations of graph theory which
shall be used in this thesis.
We define a graph G as a pair (V,E) where V is a nonempty finite set and E
is a collection of 2-subsets of V . An element of V and E is called a vertex and an
edge, respectively. Given a graph G, we denote by V (G) the set of vertices of G
and by E(G) the set of edges of G. An edge {x, y} is denoted by xy for simplicity
and the vertices x and y are called the ends point or ends of the edge. The ends of
an edge are said to be incident with the edge, and vice versa. Two vertices that are
incident with a common edge are adjacent, so are two edges that are incident with
a common vertex. A graph is trivial if it has only one vertex and empty edge set.
A graph is nontrivial if the graph is not trivial. A complete graph is a graph such
that every pair of distinct vertices is adjacent. A line graph L(G) of a graph G is
a graph such that V (L(G)) = E(G) and two vertices of L(G) are adjacent if they
have an end in common in G.
1
The degree of a vertex v in G is the number of edges of G which are incident to
v. The minimum (resp. maximum) degree of a graphG is the smallest (resp. largest)
degree of vertices in G. A graph is k-regular if the degree of each vertices is k. A
regular graph is a k-regular graph for some k.
A graph H is a subgraph of a graph G if V (H) ⊆ V (G) and E(H) ⊆ E(G).
The induced subgraph of G, induced by a nonempty subset S of V (G) denoted by
G[S], has S as its vertex set and the set of edges in G both of whose ends are in S
as its edge set. A subgraph H is a spanning subgraph of G if V (G) = V (H).
A factor T of a graph G is a spanning subgraph of G. A factor T is called
an F -factor of G if the components of T are some graphs in given a collection
F = {H1, . . . , Hn} of subgraphs of G. A matching is a set of edges no two of
which shares an end.
For a graph G, a set C ⊆ V (G) is a vertex cover of a graph G if each edge in
G has at least one end in C. The vertex cover number of a graph is the size of a
minimum vertex cover in the graph.
A complement Gc of a graph G is defined to be the graph with V (Gc) = V (G)
and for any u, v ∈ V (G), u 6= v, uv ∈ E(Gc) if and only if uv /∈ E(G).
For k ≥ 2, a graph G is called k-partite graph if there is a collection of
nonempty subsets {X1, . . . , Xk} of V (G) such that V (G) = ∪k i=1Xi and each edge
joins a vertex of Xi with a vertex of Xj for some i 6= j. A bipartite graph is
a 2-partite graph and (X1, X2) is called a bipartition of the graph. We mean by
G = (X, Y ) a bipartite graph G with bipartition (X, Y ). A complete k-partite
graph is a k-partite graph such that there is an edge vivj for each vi ∈ Xi and
vj ∈ Xj .
Let G be a graph. A walk of G is an alternating sequence of vertices and
edges v1, e1, v2, e2, . . ., vt, et, vt+1 where each vi is a vertex of G and each ei
is the edge vivi+1. A path is a walk in which all vertices are distinct. A walk
v1,e1,v2,e2,. . .,vt,et,vt+1 with at least one edge is a cycle if vt+1 = v1 and v1, v2, . . . , vt
2
are distinct. The length of path, walk, and cycle is the number of edges on the path,
walk, and cycle, respectively. The distance of two vertices v and w in a connected
graph G is the length of the shortest path from v to w. We call a cycle of length
three triangle. We say a graph is connected if there is a path between every pair of
vertices. A graph is acyclic if the graph contains no cycle as a subgraph.
For given a graph G with the vertex set V (G) = {v1, . . . , vn}, the adjacent
matrix A of G is an n × n (0, 1)-matrix such that Aij = 1 if and only if vi and vj
are adjacent. For given a bipartite graph G = (X, Y ) with X = {x1, . . . , xm} and
Y = {y1, . . . , yn}, the reduced adjacent matrix B of G is an m × n (0, 1)-matrix
such that Bij = 1 if and only if xi and yj are adjacent.
A graph G is a tree if G satisfies two conditions of the following:
1. G is connected.
2. G has no subgraph which is a cycle.
We call a vertex is a pendant vertex if the degree of the vertex is one. A caterpil-
lar is a tree such that the removal of its pendant vertices results in a path. A lobster
is a tree such that removal of pendant vertices results in a caterpillar.
A tree decomposition of a graph G is a pair ({Xp | p ∈ I}, T ) such that each
Xp, called a bag, is a subset of V (G), and T is a tree with V (T ) = I such that
• for each v ∈ V (G), there is p ∈ I with v ∈ Xp;
• for each uv ∈ E(G), there is p ∈ I with u, v ∈ Xp;
• for p, q, r ∈ I , if q is on the (p, r)-path in T , then Xp ∩Xr ⊆ Xq.
The width of a tree decomposition is the size of a largest bag minus 1. The treewidth
of a graph is the minimum width over all tree decompositions of G.
A rooted tree is a tree in which a special vertex, called the root, is singled out.
The depth of a rooted tree is the maximum number of edges of a path from a vertex
3
of the tree to the root. The tree-depth of a connected graph G is the minimum depth
of a rooted tree such that T ∗ contains G as a subgraph, where T ∗ is the subgraph of
T with the additional edges connecting all comparable pairs in T .
A planar graph is a graph which can be drawn in the plane so that no edges
cross each other. An outerplanar graph, or outplanar graph in short, is a graph that
has a planar graph such that all vertices belong to the outer face of the drawing.
A graph union G ∪H of two graphs G and H is the graph having its vertex set
V (G) ∪ V (H) and edge set E(G) ∪ E(H). A p copies of a graph G pG is a graph
∪p i=1Gi whereGi = G for 1 ≤ i ≤ p. A graph joinG∨H of two graphsG andH is
the graph having its vertex set V (G)∪V (H) and edge set E(G)∪E(H)∪E(GH)
where E(GH) is the collection of edges one of whose ends in V (G) and the other
ends in V (H). A Cartesian productG1G2 of two graphsG1 andG2 is a graph with
vertex set V (G1)× V (G2) and having two vertices (u1, u2) and (v1, v2) adjacent if
and only if either u1 = v1 and u2 is adjacent to v2 in G2, or u2 = v2 and u1 is
adjacent to v1 in G1.
An algorithm is said to be of polynomial time if its running time is upper bounded
by a polynomial expression in the size of the input for the algorithm. A decision
problem is NP if it can be solved by a non-deterministic Turing machine that runs
in polynomial time. A decision problem H is NP-hard if any NP problem can be
reduced to H in a polynomial-time. A decision problem is NP-complete if it is both
NP and NP-hard.
Other definitions will be given as needed throughout this thesis. In the follow-
ing, we list the notations used most in this thesis.
4
E(G) : The edge set of a graph G
e = uv in G : The edge e between a vertex u and a vertex v in a graph G
degG(v) : The degree of v in a graph G
G = (X, Y ) : A bipartite graph G with bipartitions X and Y
A(G) : The adjacent matrix of G
Gc : The complement of a graph G
G+ e : The graph with vertex set V (G) and edge set E(G) ∪ {e}
G− e : The graph with vertex set V (G) and edge set E(G) \ {e}
G ∪H : The union of two graphs G and H
G ∨H : The join of two graph G and H
GH : The Cartesian product of two graphs G and H
pG : The p copies of a graph G
Pn : A path of length n
Kn : A complete graph on n vertices
Km,n : A complete bipartite graph on m and n vertices
Cn : A cycle of length n
[n] : The set {1, . . . , n}
5
1.2.1 A brief history of magic labelings
The notion of magic labeling is introduced by Sedlacek [54] motivated by the notion
of magic squares in Number Theory. A magic square is an arrangement of distinct
integers in a square grid such that in each row, each column, and each diagonal, the
elements are added up to the same number. A vertex (resp. edge) labeling of a graph
G = (V,E) is a function f : V → L (resp. f : E → L) where L is a subset of the
set of integers. A total labeling of a graph G = (V,E) is a function h : V ∪E →M
whereM is a subset of the set of integers. He defined a magic labeling of a graph as
a real valued edge-labeling such that distinct edges have distinct nonnegative labels
and the sum of labels of the edges incident to a particular vertex is the same for all
vertices.
Since Sedlacek [54] introduced the notion of magic labeling of a graph in 1963,
a variety of magic labelings of a graph have been defined and studied. In mid 1966,
Stewart [50] defined the notion of semi-magic as follows:
Definition 1.2.1 ( [50]). Let G be a graph and a be an edge labeling of G. For each
vertex vi in G, the vertex sum σ(vi) := ∑i a(e) where the sum
∑i extends over all
edges of G have vi as an end point. The vertex sum satisfies semi-magic condition
if
σ(vi) = α
for any i, 1 ≤ i ≤ n, i.e., there is a ‘constant vertex sum α’ for all the vertices of G.
A semi-magic labeling is magic labeling if the labels of edges are all distinct
nonnegative integers. A magic labeling is called supermagic if the set of labels of
edges is a set of consecutive positive integers. Jeurissen [29] called a magic label-
ing positive magic if the labels of edges are positive integers to distinguish it from
6
2
3
Figure 1.1: A magic graph G and its magic labeling
a magic labeling. He provided necessary conditions and sufficient conditions for a
connected bipartite graph being magic graph. He further provided necessary condi-
tions and sufficient conditions for a connected non-bipartite graph being magic. As
the notion of magic labeling is motivated by magic squares, it is natural to consider
a positive magic labeling. In this context, positive magic labelings have been stud-
ied and variants of magic labeling also take only positive integers as labels. We say
that a graph is semi-magic, magic, or supermagic if the graph yields a semi-magic
labeling, magic labeling, or supermagic labeling, respectively. See Figure 1.1 for
an illustration. We can see that the sum of labels of edges incident to a vertex is 16
for any vertex in the graph.
There have been studies characterizing magic graphs in terms of F -factor and
matching. In 1979, Mulbacher [42] found some sufficient conditions and necessary
conditions for a graph being magic or semi-magic in terms of F -factor and match-
ing. In 1983, Jezny and Trenkler [32] provided a necessary and sufficient conditions
for a graph being semi-magic. They showed that a graph G is semi-magic if and
only if G has (1 − 2)-factor. A (1 − 2)-factor of a graph G is a factor of whose
components are an edge or a cycle. They also showed that a graph G is magic if
and only if G has (1 − 2)-factor and every couple of edges can be separated by a
7
(1− 2)-factor. Later, Trankler [57] developed his previous results. In particular, he
characterized Cartesian product and Graph join of two graphs such as cycle, path,
or triangle in terms of magic graph.
In 2010, Bezogova and Ivanco [8] introduced the notion of degree-magic. A
graph G is a degree-magic if there is an edge labeling f : E(G) → {1, . . . , |E(G)|}
such that ∑
f(e) = 1+|E(G)| 2
deg(v) for all v ∈ V (G) where the summation is taken
over all edges incident to v. They showed that a regular graph is supermagic if and
only if it is degree-magic and found a sufficient condition and a necessary condition
for a graph being a degree-magic. They also showed sufficient and necessary con-
ditions for a complete bipartite graph and a complete tripartite graph being degree
magics (see [9]). In addition, they provided a bound on the number of edges of a
degree-magic graph in terms of the number of vertices of the graph (see [10]).
There have been efforts at characterizing a specific family of magic graphs. In
1988, Doob [14] characterized regular magic graphs. He provided three equivalent
conditions for a regular connected graph being magic by using Matroid. He also
extended his results to disconnected regular graphs. He found a necessary and suf-
ficient condition for a regular graph being magic in terms of separability by even
circuits. A graph is separable by even circuits if for any pair edges, there is an
even circuit that contain exactly one of them. In 2007, Ivanco [26] showed that a
bipartite graph with high minimum degree is magic and a regular bipartite graph
with high minimum degree is supermagic. Later, Ivanco, Kova, and Semanicova-
Fenovcova [28] showed that for any graph G, there is a supermagic regular graph
which contains G as an induced subgraph.
In 2006, Semanicova [52] characterize magic circulant graphs and 3-regular su-
permagic circulant graphs. For given positive integers n,m, a1, . . . , am, a circulant
8
V (Cn(a1, . . . , am)) = {v1, . . . , vn}
E(Cn(a1, . . . , am)) = {vivi+aj | 1 ≤ i ≤ n, 1 ≤ j ≤ m}.
She also established some conditions for non-regular supermagic circluant graphs.
In 2004, Ivanco, Lastivkova, and Semanicova characterized magic line graphs and
described some class of supermagic line graphs of bipartite graphs.
An edge-labeling f : E(G) → {k, . . . , k+ |E(G)|−1} of a graph G is a k-edge
labeling if the sums of labels of edges incident to each vertices are the same modulo
|V (G)|. In 2014, Alikhani, Kocay, Lau, and Lee [3] provided some conditions for
a maximal outplanar graph having an k-edge labeling in terms of k and the number
of vertices. They also obtained some k-edge labelings of the maximal outplanar
graphs for k = 3, 4.
In 2010, Bertault, Miller, Pe-Roses, Feria-Puron, and Vaezpour [11] provided a
heuristic algorithm for finding a magic labelings. They interpreted a magic labeling
as a global optimization problem that minimizes the difference between the sums of
labels of edges incident to a vertex and their average over all vertices. As a result,
they presented an algorithm which gives a magic labeling to a specific family of
graphs. More results on characterizing magic labeling can be found in [5], [15], [16]
and [42] .
In the rest of this subsection, we give a brief history of the vertex-magic total
labeling and edge-magic total labeling which are derived from the notion of magic.
Since the notions of semi-magic and magic had been introduced, a variety of their
variants have been defined and studied. Vertex-magic total labeling and edge-magic
total labeling are notions studied most intensively among them. We first give a brief
history of vertex-magic total labeling.
In 1999, Miller, MacDougall, Slamin, and Wallis [44] introduced the notion of
vertex-magic total labeling.
Figure 1.2: A graph K4 − e and its vertex-magic labeling
Definition 1.2.2 ( [44]). For given a graph G, a total labeling f : V (G)∪E(G) →
{1, . . . , |V (G)| + |E(G)|} is vertex-magic total labeling if there is a constant k so
that for every vertex v ∈ V (G),
f(v) + ∑
f(vw) = k
where the sum is over all vertices w adjacent to v.
The constant k in Definition 1.2.2 is called the magic constant of the vertex-
magic total labeling or a magic constant of G. We call a graph G vertex-magic total
if it has a vertex-magic total labeling. See Figure 1.2 for an illustration. We can see
that a graph K4 − e is vertex-magic total.
In 2002, Miller, MacDougall, Slamin, and Wallis [45] provided some properties
of vertex-magic total regular graphs. They also showed that Cn and Pn are vertex-
magic total for n ≥ 3. In addition, they showed that Km,n is not vertex-magic total
if n > m + 1, Km,m is vertex-magic total for m > 1 and Kn is vertex-magic total
for all odd n and even n not divisible by 4. They posed two open conjectures: one is
10
that Kn is vertex-magic total for all n ≥ 3 and the other is that Kn has vertex-magic
total labelings for every feasible magic constants.
The former conjecture had proven very early. In 2001, Lin and Miller [41]
showed Kn is vertex-magic total for all n divisible by 4. With the result had shown
in [45], they concluded thatKn is vertex-magic total for all n ≥ 3. Later Gray, Mac-
Dougall, and Wallis [25] provided another proof for the problem by using matrix
in 2003. More recently, Krishnappa, Kothapalli, and Venkaiah [35] gave a simpler
proof for the problem by decomposing Kn into several smaller graphs.
There has been quite a lot of study on the latter conjecture. In 2005, McQuil-
lan and Smith [46] answered the latter conjecture partially. They showed that the
conjecture is true for every odd n ≥ 3. They also generalized the conjecture by
considering existence of vertex-magic labeling of pKn. Later, in [47], they showed
that pKn is vertex-magic total for every positive integer p and n ≥ 4. In 2011,
Armstrong and McQuillan [4] answered the conjecture posed by McQuillan and
Smith [46] partially by showing that 2Kn has vertex-magic total labelings for every
feasible magic constants for all odd n ≥ 5.
Besides the complete graphs, there has been studies on vertex-magic total graphs
obtained from graph operations of vertex-magic total graphs. Gray [23] provided
several ways of constructing regular vertex-magic graphs. He further introduced
ways of obtaining certain classes of non-regular graphs using graph union. In 2008,
Gomez [22] showed that if a regular graph G is vertex-magic total, then kG is
vertex-magic total. He provided the bound of magic constants of kG. Armstring
and McQuillan [4] showed that if two graphs G1 and G2 are vertex-magic total and
G1 ∪G2 is also vertex-magic, then G1 ∨G2 is vertex-magic and the magic constant
is bounded in certain interval. This results answered the latter conjecture partially.
Edge-magic total labeling is one of the popular notion derived from the magic
labeling. Motivated by the notion of magic labeling given by Sedlacek [54], Kotzig
and Rosa [39] introduced the notion of magic valuation of a graph.
11
1
1
22
3
Figure 1.3: An edge-magic labeling of C5 and P3.
Definition 1.2.3 ( [39]). A graph G with m vertices and n edges is said to have
a magic valuation (M-valuation) with the constant C if there exists a one-to-one
mapping f : V (G)∪E(G) → {1, . . . , m+n} such that f(a)+f(b)+f([a, b]) = C
for all [a, b] ∈ E(G).
The constant C in Definition 1.2.3 is called a magic constant of the magic val-
uation or a magic constant of the graph. This notion was rediscovered by Ringel
and Llado [49] in 1996. They called this labeling as edge-magic labeling. More
recently, Marr and Wallis [58] used the term edge-magic total labeling to distin-
guish it from other kinds of labelings that use the word magic. We say that a graph
is edge-magic if the graph has an edge-magic total labeling. See Figure 1.3 for an
illustration.
A variety of physical processes can be modelled by assigning integer values on
the vertices and edges of complete graphs. We obtain a ruler model from a labeling
λ ofKn by placing a mark distance λ(v) from the start for each v ∈ V (Kn). We use
the ruler to measure the distances for every pair of marks. The ruler model derived
from the edge-magic labeling on vertices has the following special property. Let
f be an edge-magic labeling of Kn with magic constant k. If there are two pairs
(v1, v2) and (w1, w2) of vertices with the same distance, then λ(v1) − λ(v2) =
12
λ(w1)− λ(w2) or λ(v1) + λ(w2) = λ(w1) + λ(v2), so k− λ(v1w2) = k − λ(w1v2)
which implies either v1 = v2 and w1 = w2 or {v1, v2} = {w1, w2}. Therefore the (
n
2
distances of Kn are all distinct.
There has been much research on studying edge-magic total labelings. In 1970,
Kotzig and Rosa [39] showed that a complete bipartite graph Kp,q is edge-magic
total for all p, q ≥ 1 and n-gon is edge-magic total for n ≥ 3. They also showed
that a tree with limited number of pendant vertices is edge-magic total. Later, Kotzig
and Rosa [40] proved that a complete bipartite graph Km,n has an edge-magic total
labeling for all m and n, a cycle Cn has an edge-magic total labeling for all n ≥
3, and nP2 has an edge-magic total labeling if and only if n is odd. They also
showed that a complete graph Kn has an edge-magic total labeling if and only if
n = 1, 2, 3, 5 or 6.
In 1975, Wallis [58] summarized results on edge-magic total labelings and pro-
vided some conditions for a graph not having edge-magic total labeling. In addition,
he gave necessary conditions for some specific graphs such as cycles, complete bi-
partite graphs, and wheels to have an edge-magic total labeling.
In 1999, Craft and Tesar [13] showed that an r-regular graph with n vertices is
not edge-magic total if n ≡ 4 (mod 8) and r is odd. They went further to show that
if a graph G is edge-magic total with magic constant C, then there is an edge-magic
total labeling of G with magic constant 3(|V (G)|+ |E(G)|+ 1)− C.
In 2000, Baskoro and Cholily [7] showed that Cn + Ap is super-edge magic for
all odd n. For n ≥ 3 and p ≥ 1, the graph Cn +Ap is a graph obtained by adding p
pendant vertices to one vertex of a cycle Cn. Baskoro and Cholily also showed that
an (n, p)-sun is super edge-magic for all odd n. An (n, p)-sun is a graph obtained
by adding p pendant vertices to each of vertices of Cn. They further showed that the
Peterson graph is super edge-magic. The Peterson graph is a graph with 10 vertices
and 15 edges.
In 2000, Wallis, Baskoro, Miller and Slamin [59] introduced some basic proper-
13
ties of an edge-magic total labeling of graphs. Then they provided edge-magic total
labelings for families of graphs such as odd cycles, suns, (n, 1)-kites, complete
bipartite graphs and stars.
In 2002, Slamin, Baca, Lin, Miller, and Simanjuntak [53] showed that a wheel
Wn = Cn−1∨K1 is edge-magic total for all n ≡ 6 (mod 8) with the magic constant
5n+2. They posed an conjecture thatWn is edge-magic total for all n ≡ 2 (mod 8).
They also showed that a fan graph fn = Pn−1∨K1 is edge-magic total for all n ≥ 2
with magic constant 4n + 2.
1.2.2 Super edge-magic labelings and Enomoto’s conjecture
In this subsection, we introduce a special type of edge-magic labeling called con-
secutive edge-magic labeling and Enomoto’s conjecture. Motivated by the notion
of magic valuation, Enomoto, Llado, Nakamigawa, and Ringel [17] introduced the
notion of super-edge magic total labeling.
Definition 1.2.4 ( [17]). An edge-magic total labeling f of a graphG is super edge-
magic total labeling if
and
They provided a simple, but very useful lemma.
Lemma 1.2.5 ( [17]). If G is a super edge-magic, then |E(G)| ≤ 2|V (G)| − 3.
They showed that a cycle Cn is super edge-magic if and only if n is odd, a
complete bipartite graph Km,n is super edge-magic if and only if m = 1 or n = 1
and a wheel graph Wn is not super edge-magic.
14
After the notion has been introduced, Figueroa-Centeno, Ichishima, and Muntaner-
Batle [18] presented the following useful lemma.
Lemma 1.2.6 ( [18]). A graph G is super edge-magic if and only if there exists a
bijective function f : V (G) → {1, . . . , p} such that the set
S = {f(u) + f(v) | uv ∈ E(G)}
consists of |E(G)| consecutive integers. In such a case, f extends to a super edge-
magic labeling of G with magic constant k = |V (G)| + |E(G)| + s, where s =
min(S) and
S = {k − (|V (G)|+ 1), k − (|V (G)|+ 2), . . . , k − (|V (G)|+ |E(G)|)}.
They investigated on the relationship between the magic constants of a super
edge-magic graph and the number of vertices and edges. They also showed that
the fan graph fn is super edge-magic if and only if 1 ≤ n ≤ 6, the ladder graph
Pn×P2 is super edge-magic if n is odd and the generalized prism Cm×Pn is super
edge-magic if m is odd and n ≥ 2.
Since the notion has been introduced, there has been research finding super
edge-magic total labelings for specific families of graph. Especially, finding a super-
edge magic total labeling of a graph obtained by operating well-known super-edge
magic graphs has been widely studied.
In 2005, Figuero-Centeno, Ichishima, and Muntaner-Batle [19] showed the fol-
lowing. If a super edge-magic graph G is bipartite or tripartite, then mG is super
edge-magic for odd m; m is a multiple of n+ 1 if and only if K1,m ∪K1,n is super
edge-magic; F := K1,2 ∪K1,n is super edge-magic if and only if n is multiple of 3.
As a matter of fact, there are essentially two super edge-magic labeling of F ; For
m ≥ 4 and n ≥ 1, Pm ∪K1,n is super edge-magic; 2Pn is super edge-magic if and
15
only if n 6= 2 or 3. K1,m ∪ 2nP2 is super edge-magic; If a super edge-magic graph
G with |V (G)| ≥ 4 and |E(G)| ≥ 2|V (G)| − 4, then G contains a triangle.
In 2011, Figuero-Centeno, Ichishima, Muntaner-Batle, and Oshima [20] ob-
tained the following results. The m copies of Cn is super edge-magic if and only if
both of m and n are odd; the union of C3 and Cn is super edge-magic if and only if
n ≥ 6 and n is even; C4∪Cn is super edge-magic if and only if n ≥ 5 and n is odd;
C5 ∪Cn is super edge-magic if and only if n ≥ 4 and n is even; For an even m ≥ 4
and odd n with n ≥ m/2 + 2, Cm ∪Cn is super edge-magic; C3 ∪Pn is super-edge
magic for n ≥ 6; C4∪Pn is super edge-magic if and only if n 6= 3; C5∪Pn is super
edge-magic for n ≥ 4; Cm ∪ Pn is super edge-magic if m is even with m ≥ 4 and
n ≥ m/2 + 2; For every n ≥ 3, P2 ∪ Pn is super edge-magic; for n ≥ 4, P3 ∪ Pn
is super edge-magic; Pm ∪ Pn is super edge-magic if and only if (m,n) 6= (2, 2) or
(3, 3).
In 2014, Afzal, Baig, Imran, and Javaid [1] showed that a volvox graph that the
order of cycle is even and the number of cycle is odd is super edge-magic. A volvox
graph is a graph obtained by adding chords and edges to disjoint union of cycles of
the same order which describing the colony of volvox, which is one of the famous
chlorophyte. They also showed that a pancyclic graph is super edge-magic. The
pancyclic graph on n vertices is a graph having cycles of length 3, . . . , n.
One of the popular topics on the research on edge-magic total labeling is an-
swering the following two conjectures. One was posed by Kotzig and Rosa [39].
Conjecture 1.2.7 ( [39]). Every tree is edge-magic.
The other was given by Enomoto et al. [17] who introduced the notion of super
edge-magic total labeling.
Conjecture 1.2.8 ( [17]). Every tree is super edge-magic.
There were various approaches to the conjectures. It is almost impossible to find
16
labelings for every specific trees. One of the approaches widely studied is obtaining
new super edge-magic trees from a family of super edge-magic trees.
A w-graph W (n) has vertex set {c1, c2, b, w, d} ∪ {x1, . . . , xn} ∪ {y1, . . . , yn}
and edges {c1x1, . . . , c1xn}∪{c2y1, . . . , c2yn}∪{c1b, c1w}∪{c2w, c2d}. A w-tree
WT (n, k) is a tree obtained by taking k copies of W (n) and joining a new vertex
a with each copy of d. In 2011, Javaid, Hussain, Ali, and Dar [30] showed that for
k ≥ 3, WT (n, k) is super edge-magic if n ≥ k− 1 for even k and n ≥ k for odd k.
We denote by T (n1, n2, n3, n4) the subdivision of K1,4 obtained by inserting ni − 1
vertices into each edges of K1,4 for i = 1, 2, 3, 4. In 2012, Javaid, Hussain, Ali, and
Shaker [31] showed that T (n, n, n, n) and T (n, n, n−1, n) are super edge-magic for
any odd n ≥ 3. Then they showed that for n ≥ 2 and k ≥ 3, subdivided tree w-tree
WT (n, k) is super edge-magic for n ≥ 2k−2. In 2015, to generalize T (n, n, n, n),
Ali, Hussain, Shaker, and Javaid [2] introduced the graph T (n1, n2, . . . , np) which
is a graph obtained by inserting ni − 1 vertices to each of ith edge of the star K1,p
for 1 ≤ i ≤ p. They showed that T (n, n, n − 1, 2n − 1) is super edge-magic for
any odd n ≥ 3; T (n, n, n − 1, n, 2n− 1, 4n− 3) is super edge-magic for any odd
n ≥ 3; T (n, n, n − 1, n, n5, . . . , np) is super edge-magic for any odd n ≥ 3 and
p ≥ 5 where n5 = · · · = np = n + (n− 1)(p− 3)(p− 4)/2.
Let G1, . . . , Gn be disjoint stars and vi be a pendant vertex of Gi for 1 ≤ i ≤ n.
A fire cracker is the tree which contains all n stars and a path joining v1, . . . , vn.
For a vertex no in any of G1, . . ., Gn, a banana tree is the tree obtained by joining v
to one pendant vertex of each star. In 2006, Swaminathan and Jeyanthi [55] showed
that a fire cracker is super edge-magic and enumerated the bound of the minimum
magic constant and that a banana tree is super edge magic and enumerated the bound
of the minimum magic constant.
Baskoro, Sudarsana, and Cholily [12] designed algorithms to construct new su-
per edge-magic graphs by extending the old ones. They obtained new super edge-
magic graph by adding one, two, three or |V (G)| pendant vertices to a super edge-
17
magic graph. In this way, they found a new super edge-magic tree from a super
edge-magic tree.
Fukuchi and Oshima [21] showed that a tree with n vertices is a super edge-
magic if it has diameter greater than or equal to n − 5. Later in 2013, Oshima and
Sawaki [48] improved the result by soothing the diameter condition to n− 6.
1.2.3 b-edge consecutive magic labelings
A super vertex-magic labeling and super edge-magic labeling total has a special
property in common that the labels on the vertices and the edges are consecutive
integers. In this subsection, we introduce one of the interesting approach to the
topic which focused on the property mentioned above.
Balbuena, Barker, Lin, Miller, and Sugeng [6] introduced notions on vertex-
magic total labeling which generalized the notion of super vertex-magic total label-
ing.
Definition 1.2.9 ( [6]). A vertex-magic total labeling λ of a graph G is said to be
a-vertex consecutive if
λ(V (G)) = {a+ 1, . . . , a+ |V (G)|}
for some a ∈ {0, . . . , |E(G)|}. Analogously, λ is said to be b-edge consecutive if
λ(E(G)) = {b+ 1, . . . , b+ |E(G)|}
for some b ∈ {0, . . . , |V (G)|}.
We can easily see that a super vertex-magic total labeling is a special type of
a-vertex consecutive and b-edge consecutive.
In their paper, they introduced many properties of vertex-magic total labeling
and edge-magic total labeling. They gave an equality on the number of vertices and
18
edges of an a-vertex consecutive graph with one isolated vertex. They showed that
if G is an a-vertex consecutive graph with the number of edges is one less than the
number of vertices, then the number of vertices is odd and a equals the number of
edges. They provided the feasible values of a in terms of the number of vertices and
the number of edges when an a-vertex consecutive graph with the minimum degree
at least two or three is given.
For a b-edge consecutive magic graph G having n vertices and e edges, they
showed that ifG has one isolated vertex, then b = 0 and the equality (n−1)2+n2 =
(2e + 1)2 holds; if e = n − 1 and n is odd, then b = 0; if e = n− 1 and n is even,
then b = n 2 ; if G is an even tree, then G is a path of length 4; If e ≥ 3n/2 and
b ≥ n/4, then the minimum degree of G is greater than 2; If G is 2-regular, then
b = 0 or b = n; If G is r-regular, then n and r have distinct parities.
In 2015, Marimuthu and Kumar [37] gave a lower bound of magic constants
of an a-vertex consecutive magic total labelings of connected graph and an upper
bound of magic constants a-vertex consecutive magic total labelings of connected
graph. They also obtained a lower bound of magic constants of an a-vertex consec-
utive magic total labelings of graphs without isolated vertices and an upper bound
of magic constants of magic constants of a-vertex consecutive magic total labeling.
Later, Sugeng and Miller [56] applied the notions to edge-magic labeling simi-
larly.
Definition 1.2.10 ( [56]). An edge-magic total labeling λ of a graph G is said to be
a-vertex consecutive if
λ(V (G)) = {a+ 1, . . . , a+ |V (G)|}
for some a ∈ {0, . . . , |E(G)|}. Analogously, λ is said to be b-edge consecutive if
λ(E(G)) = {b+ 1, . . . , b+ |E(G)|}
for some b ∈ {0, . . . , |V (G)|}.
19
A graph which admits a b-edge consecutive magic labeling is said to be b-edge
consecutive magic.
In [56], Sugeng and Miller studied on a b-edge consecutive magic labeling
which focused on b which is neither 0 nor the number of vertices. They showed
that for every b, there is a caterpillar which is b-edge consecutive magic label-
ing and every caterpillar is b-edge consecutive magic labeling for some b. They
showed that a connected graph G is a tree if G is b-edge consecutive magic for
some 1 ≤ b ≤ |V (G)| − 2. In addition, they showed a graph consist of r union of
stars with r − 1 isolated vertices is an r-edge consecutive magic.
1.3 Safe sets of graph
We first introduce the definition of a safe set and a connected safe set.
Definition 1.3.1. For a connected graph G, a set S of vertices in G is said to be a
safe set if for every component C of the subgraph induced by S, |C| ≥ |D| holds for
every component D of G \ S such that there exists an edge between C and D, and,
especially, if the subgraph induced by S is connected, then S is called a connected
safe set.
For a connected graph G, the safe number s(G) of G is defined as s(G) =
min{|S| | S is a safe set of G}, and the connected safe number cs(G) of G is de-
fined as cs(G) = min{|S| | S is a connected safe set of G}.
The notions of safe set and connected safe set are motivated by the problem
given in Fujita et al. [61]. They first focused on the relation between the safe number
and connected safe number. They showed that for a graph G, s(G) ≤ cs(G) ≤
2s(G) − 1, any tree T with at most one vertex of degree at least three satisfies the
equality s(T ) = cs(T ) and any tree with n vertices has a connected safe set of size
20
v1
v2
v3
v4
v5
v6
G
Figure 1.4: A set {v4, v5, v9, v11} is a safe set. However, S = {v3, v4, v9, v11} is
not a safe set as G− S has the component with 4 vertices v5, v6, v7, v8 even if each
component of the subgraph of G induced by S has size two.
⌊n 3 ⌋. Then they focused on the problem of finding a safe set and a connected safe
set of size less than given integer. They turned the problem into decision problems
and showed that determining the existence of a safe set (resp. a connected safe set)
of a graph of size less than given integer is NP-complete. Yet they found a sharp
upper bound of the safe number of a tree and a polynomial-time algorithm which
gives the connected safe number of a tree.
Bapat et al. [60] generalized the notion of safe set by considering weights of
vertices. Let G be a graph and w be a positive weighted function on V (G) such that
w : V (G) → Q+ where Q+ is the set of positive rational numbers. For a vertex
subset S of V (G), let w(S) = ∑
v∈S w(v). For a subgraph H of G, w(H) stands
for w(V (H)). The following are the definitions of weighted safe set and weighted
connected safe set.
Definition 1.3.2. Let G be a graph and w : V (G) → Q+ be a positive weight
function on vertices of G. A non-empty subset S ⊂ V (G) is a weighted safe set if,
for every component C of the subgraph induced by S and every component D of
21
G \ S, we have w(C) ≥ w(D) whenever there is an edge joining a vertex of C and
a vertex of D in G. If G[S] is connected, then S is called a weighted connected safe
set.
The minimum weight among all weighted safe sets (resp. connected safe sets) of
(G,w) is called the safe number (resp. the connected safe number) of (G,w) and is
denoted by s(G,w) (resp. cs(G,w)).
They investigated the problem of finding a weighted safe set and a weighted
connected safe set of a graph of size less than given integer. As Fujita et al. [61]
did, they turned the problem into decision problems and showed that determining
the existence of a weighted safe set (resp. a weighted connected safe set) of a graph
of size less than given integer is NP-complete even for stars. They showed that
finding a safe set of minimum weight of a path is solvable in a polynomial-time and
that all safe sets of minimum weight of a path can be enumerated with polynomial-
time delay.
Later, Agueda et al. [62] designed a polynomial-time algorithm of finding a safe
set of a tree. Then they extended their results to present a polynomial algorithm of
finding safe sets of graphs with bounded treewidth and safe sets of interval graphs.
They showed that a class of graphs with constant safe number has a bounded tree-
depth and that the safe number of a graph is at most its vertex cover number. They
also showed that for a vertex weighted graph of bounded treewidth, a weighted
safe set and a weighted connected safe set of the minimum weight can be found in
pseudo-polynomial time.
1.4 A preview of thesis
Study on consecutive edge-magic labelings is one of the most popular topic on the
study of graph labeling. Especially, characterizing consecutive edge-magic graphs
22
is one of the most popular topic in the study of consecutive edge-magic labeling.
Though there has been much study on the topic, determining whether or not it yields
a consecutive edge-magic labeling remains open even for graphs with rather simple
structures. In this thesis, we study properties of consecutive edge-magic graphs and
construct new consecutive edge-magic graph from existing consecutive edge-magic
graphs through graph operation. Then we provide several ways of obtaining new
consecutive edge-magic graphs from consecutive edge-magic graphs through graph
operation.
This thesis is organized into three chapters. In the first chapter, we give some ba-
sic terminology and results on consecutive edge-magic graphs and safe sets. In the
second chapter, we give properties of consecutive edge-magic graphs and provide
several ways of obtaining new consecutive edge-magic graphs from consecutive
edge-magic graphs through graph operation. In the third chapter, we design an al-
gorithm which gives the safe number and the connected safe number of Cartesian
product of two complete graphs.
In Chapter 2, we give properties of consecutive edge-magic graphs and provide
several ways of obtaining new consecutive edge-magic graphs from consecutive
edge-magic graphs through graph operation. To put it concretely, we show that a
graph G admitting b-edge consecutive magic labeling for 0 < b < |V (G)| is a
graceful tree having super edge-magic labeling. Then we provide several ways of
obtaining new consecutive edge-magic graphs from consecutive edge-magic graphs
through graph operation. Especially, we show that there are infinitely many consec-
utive edge-magic trees containing arbitrary number of copies of given a consecutive
edge-magic tree. In addition, we give bounds for the number of consecutive edge-
magic graphs paired with its consecutive edge-magic labelings through the corre-
spondence between the pairs and the adjacent matrices of the graphs having special
properties.
In Chapter 3, we study the safe number and the connected safe number of
23
Cartesian product of two complete graphs. First, we analyze the structure of Carte-
sian product of two complete graphs and present ways of restricting the minimum
safe set and the minimum connected safe set of Cartesian product of two complete
graphs to simple forms. Then we design a polynomial-time algorithm of comput-
ing the safe number and the connected safe number of Cartesian product of two
complete graphs. Based on these results, we show that the safe number and the
connected safe number of Cartesian product of two complete graphs are equal.
24
labeling of a tree
bipartite graphs 1
2.1.1 Properties of edge-magic total labelings
We first present the following proposition which is simple but very useful.
Proposition 2.1.1. Suppose that G has a b-edge consecutive magic labeling λ for
1 ≤ b ≤ |V (G)|. Then, if y and z are neighbors of a vertex x, then either
{λ(y), λ(z)} ⊂ {1, . . . , b} or {λ(y), λ(z)} ⊂ {b + |E(G)| + 1, . . . , |V (G)| +
1The material in the section is from the paper “On consecutive edge magic total labelings of
connected bipartite graphs” by Bumtle Kang, Suh-Ryung Kim, and Ji Yeon Park, which will appear
in Journal of Combinatorial Optimization. Thus auther thanks Professor Suh-Ryung Kim and Dr. Ji
Yeon Park for allowing him to use its contents for his thesis.
25
|E(G)|}.
Proof. Without loss of generality, we may assume that λ(y) < λ(z). Suppose to
the contrary that λ(z) > b and λ(y) < b + |E(G)| + 1. Then, by the definition of
b-edge consecutive magic labeling, λ(z) ≥ b + |E(G)| + 1 and λ(y) ≤ b. Since λ
is an edge-magic total labeling,
λ(x) + λ(y) + λ(xy) = λ(x) + λ(z) + λ(xz),
which implies
λ(y) + λ(xy) = λ(z) + λ(xz) (2.1)
Since λ is a b-edge consecutive magic labeling, λ(xz) ≥ b+ 1. From (2.1) and the
assumption that λ(z) ≥ b+ |E(G)|+ 1, we obtain
λ(y) + λ(xy) ≥ (b+ |E(G)|+ 1) + (b+ 1) = 2b+ |E(G)|+ 2.
Again, by the definition of b-edge consecutive magic labeling, λ(xy) ≤ b+ |E(G)|.
Thus we have
which contradicts the assumption that λ(y) ≤ b.
The following theorem shows that there are only four possible values of b for
which a connected bipartite graph has a b-edge consecutive magic labeling.
Theorem 2.1.2. If a connected bipartite graph G = (X, Y ) has a b-edge consecu-
tive magic labeling, then b ∈ {0, |X|, |Y |, |X|+ |Y |}.
Proof. For simplicity, we denote the set {1, 2, . . . , b} by [b]. By the definition of
a b-edge consecutive magic labeling, it is sufficient to show that b = |X| or |Y |
26
if 1 ≤ b < |X| + |Y |. Suppose that 1 ≤ b < |X| + |Y |. We claim that either
λ(X) ⊂ [b] or λ(X) ⊂ {b + |E(G)| + 1, . . . , |X| + |Y | + |E(G)|}. Take two
vertices x and x′ in X . Since G is connected, there is an (x, x′)-path P in G. Since
G is bipartite,
P = x1w1x2w2x3 · · ·xlwlx ′
where x = x1 and xi ∈ X and wi ∈ Y for i = 1, . . ., l where l is a positive integer.
Then, by Proposition 2.1.1, if λ(x) ≤ b, then λ(xi) ≤ b for i = 2, . . ., l and so
λ(x′) ≤ b, and if λ(x) ≥ b+ |E(G)|+1, then λ(xi) ≥ b+ |E(G)|+1 for i = 2, . . .,
l and so λ(x′) ≥ b+ |E(G)|+1. Since x and x′ were arbitrarily chosen, λ(X) ⊂ [b]
or λ(X) ⊂ {b+ |E(G)|+1, . . . , |X|+ |Y |+ |E(G)|}. Similarly we may show that
λ(Y ) ⊂ [b] or λ(Y ) ⊂ {b+ |E(G)|+ 1, . . . , |X|+ |Y |+ |E(G)|}.
If λ(X) ⊂ [b] and λ(Y ) ⊂ [b], then b ≥ |X| + |Y |, which contradicts the
assumption. Suppose that λ(X) ⊂ {b + |E(G)| + 1, . . . , |X| + |Y | + |E(G)|}
and λ(Y ) ⊂ {b + |E(G)| + 1, . . . , |X| + |Y | + |E(G)|}. Since the size of the set
{b+ |E(G)|+ 1, . . . , |X|+ |Y |+ |E(G)|} is |X|+ |Y | − b, we have |X|+ |Y | ≤
|X| + |Y | − b. Therefore b = 0 and we reach a contradiction to the assumption
again. If λ(X) ⊂ [b] and λ(Y ) ⊂ {b + |E(G)| + 1, . . . , |X| + |Y | + |E(G)|},
then |X| ≤ b and |Y | ≤ |X| + |Y | − b, which implies b = |X|. If λ(X) ⊂
{b+ |E(G)|+1, . . . , |X|+ |Y |+ |E(G)|} and λ(Y ) ⊂ [b], then |X| ≤ |X|+ |Y |−b
and |Y | ≤ b, which implies b = |Y |.
Remark 2.1.3. A caterpillar is a tree derived from a path by joining leaves to the
vertices of the path. We denote by Sn1,n2,...,nr the caterpillar derived from a path
Pr = c1c2 · · · cr for a positive integer r by joining ni leaves to ci, where ni is a
nonnegative integer, for each i = 1, . . ., r. We denote the neighbors of ci by ci,1, . . .,
ci,ni for i = 1, . . ., r. Sugeng and Miller [56] gave the following theorem:
27
Theorem 2.1.4 ( [56]). Every caterpillar has a b-edge consecutive edge magic la-
beling, where
⌈
⌉
+ ∑
i even,i<r ni − 2 + (nr − 1) if r is even.
Theorem 2.1.4 can be rephrased as follows: A caterpillar Sn1,n2,...,nr has a b-
edge consecutive magic labeling if
b ∈ {0,
⌈ r 2 ⌉
i=1
ni + r}.
Therefore Theorem 2.1.2 guarantees that the converse of Theorem 2.1.4 is also true.
Since the double star Sm,n is a special case of the caterpillar Sn1,n2,...,nr for
r = 2, the following is true:
The double star Sm,n has a b-edge consecutive magic labeling if and
only if b ∈ {0, m+ 1, n+ 1, m+ n + 2}.
Suppose that a graph G with m edges has a labeling of its vertices with some
subset of {0, 1, . . . , m} such that no two vertices share a label and the edge labels
are the set {1, 2, . . . , m} where an edge label is the difference of the values assigned
to its end vertices. Then G is said to be graceful and such a labeling is called a
graceful labeling of G.
A b-edge consecutive magic labeling of a graph is called a super edge-magic
labeling if b = |V (G)|. A graph G is said to be super edge-magic if it has a super
edge-magic labeling. Enomoto et al. [17] conjectured that every tree is super edge-
magic, which still remains open.
We obtain an interesting result related to graceful labeling and super edge-magic
labeling. Furthermore, the conjecture given in [17] is true for a tree T that has an
28
|X|-edge consecutive magic labeling where X is one of bipartitions of T when it is
considered as a bipartite graph.
We first show the following:
Theorem 2.1.5. If a connected non-bipartite graph G has a b-edge consecutive
magic labeling, then b = 0 or |V (G)|.
Proof. It suffices to show that if b ≥ 1, then b = |V (G)|. Suppose that b ≥ 1.
Then there is a vertex x in G such that λ(x) ≤ b. Take a vertex y in G. Since
G is not a bipartite graph, there is an odd cycle C in G. Let w be a vertex on C.
Since G is connected, there is an (x, w)-path P1 and a (w, y)-path P2 in G. Let W
be a walk which is obtained by concatenating P1 and P2 if the sum of lengths of
P1 and P2 is even, and by concatenating P1, C, and P2 if the sum of lengths of P1
and P2 is odd. In both cases, the walk W is an (x, y)-walk of even length. Then
W = v1v2 · · · v2lv2l+1 for some positive integer l where v1 = x and v2l+1 = y. By
Proposition 2.1.1, λ(v3) ≤ b since λ(x) ≤ b. Then, by repeatedly applying the same
proposition, we have λ(v2l+1) ≤ b, that is, λ(y) ≤ b. Since y was arbitrarily chosen,
λ(v) ≤ b for any v ∈ V (G). Thus |V (G)| ≤ b, and therefore |V (G)| = b.
Let G be a connected graph with n vertices having a b-edge consecutive magic
labeling for some b ∈ [n − 1]. Then, by the above theorem, G is bipartite. If
b = n− 1, then one of the partite sets of G has size n− 1 by Theorem 2.1.2 and so
G is a star. Hence the theorem below by Sugeng and Miller [56] may be extended
to include the case b = n− 1:
Theorem 2.1.6 ( [56]). If a connected graph G with n vertices has a b-edge con-
secutive magic labeling with 1 ≤ b ≤ n− 2, then G is a tree.
The following theorem shows that G should be a special tree that has both a
graceful labeling and a super edge-magic labeling.
29
Theorem 2.1.7. If a connected bipartite graph G = (X, Y ) has an |X|-edge con-
secutive magic labeling, thenG is a tree having both a graceful labeling and a super
edge-magic labeling.
Proof. We first show that G has a graceful labeling. Let λ be an |X|-edge consec-
utive magic labeling of G. Then, by Proposition 2.1.1, either λ(X) = {1, . . . , |X|}
or λ(Y ) = {1, . . . , |X|} under the condition |X| = |Y |. By symmetry, we may
assume that λ(X) = {1, . . . , |X|}. Then, by the definition of |X|-edge consecutive
magic labeling, λ(Y ) = {|X|+ |E(G)|+ 1, . . . , |X|+ |Y |+ |E(G)|}. Therefore
λ(X) = {1, . . . , |X|}, λ(Y ) = {|X|+ |E(G)|+ 1, . . . , |X|+ |E(G)|+ |Y |},
(2.2)
and
λ(E(G)) = {|X|+ 1, . . . , |X|+ |E(G)|}.
Since |λ(E(G))| = |E(G)| and λ(x) + λ(y) + λ(xy) is constant for each edge xy
of G,
|{λ(x) + λ(y) | x ∈ X, y ∈ Y, xy ∈ E(G)}| = |E(G)|. (2.3)
We define a labeling from X ∪ Y to {0, 1, . . . , |X|+ |Y | − 1} as follows:
(z) =
|E(G)|+ 2|X|+ |Y | − λ(z) if z ∈ Y.
Then, by (2.2),
(X) = {0, 1, . . . , |X| − 1} and (Y ) = {|X|, . . . , |X|+ |Y | − 1} (2.4)
and, for a pair of adjacent vertices x ∈ X and y ∈ Y ,
1 ≤ (y)− (x) ≤ |X|+ |Y | − 1.
30
Now fix a pair of adjacent vertices x ∈ X and y ∈ Y . Since G is connected,
|X|+ |Y | − 1 ≤ |E(G)| and so
1 ≤ (y)− (x) ≤ |X|+ |Y | − 1 ≤ |E(G)|. (2.5)
Now (y) > (x) and
= |E(G)|+ 2|X|+ |Y |+ 1− (λ(x) + λ(y)). (2.6)
Thus, by (2.3) and (2.6), S := {(y) − (x) | x ∈ X, y ∈ Y, xy ∈ E(G)} has
|E(G)| elements. Therefore, by (2.5),
S = {1, 2, . . . , |E(G)|}
which, together with (2.4), implies that is a graceful labeling of G. In addition,
it follows that there is a pair of adjacent vertices x∗ ∈ X and y∗ ∈ Y such that
(y∗) − (x∗) = |E(G)|. Then, since adjacent vertices x and y are arbitrarily
chosen from X and Y , respectively, in (2.5),
|E(G)| = (y∗)− (x∗) ≤ |X|+ |Y | − 1 ≤ |E(G)|.
Thus |X|+ |Y | − 1 = |E(G)| and so we may conclude that G is a tree.
Now we show that G has a super edge-magic labeling. Let ψ be an |X|-edge
consecutive magic labeling. Then ψ(X) = {1, . . . , |X|}; ψ(E(G)) = {|X| +
1, . . . , |X|+ |E(G)|}; ψ(Y ) = {|X|+ |E(G)|+ 1, . . . , |X|+ |Y |+ |E(G)|}.
Now we define ψ∗ : X ∪ Y ∪ E(G) → {1, . . . , |X|+ |Y |+ |E(G)|} by
ψ∗(v) =
31
and, for each pair of adjacent vertices x and y,
ψ∗(xy) = ψ(xy) + |Y |.
For an edge xy of G for x ∈ X , y ∈ Y ,
ψ∗(x) + ψ∗(y) + ψ∗(xy) = ψ(x) + (ψ(y)− |E(G)|) + (ψ(xy) + |Y |),
which is a constant number. Thus ψ∗ is an edge-magic labeling of G. By the
definition of ψ∗, ψ∗ is an (|X| + |Y |)-edge consecutive magic labeling, that is, a
super edge-magic labeling.
Given a graph G, let γ : V (G) ∪ E(G) → {1, 2, . . . , |V (G)| + |E(G)|} be
an edge-magic labeling for a graph G. Define the labeling γ′ : V (G) ∪ E(G) →
{1, 2, . . . , |V (G)|+ |E(G)|} as follows: For a vertex x,
γ′(x) = |V (G)|+ |E(G)|+ 1− γ(x),
for an edge xy,
γ′(xy) = |V (G)|+ |E(G)|+ 1− γ(xy).
Then γ′ is called the dual of γ. From Marr and Wallis [58], we know that the dual of
an edge-magic labeling of a graphG is also an edge-magic labeling ofG. Moreover,
if k is the magic constant corresponding to γ, then for any adjacent vertices x and y
of G,
= (|V (G)|+ |E(G)|+ 1− γ(x)) + (|V (G)|+ |E(G)|+ 1− γ(y))
+ (|V (G)|+ |E(G)|+ 1− γ(xy))
= 3(|V (G)|+ |E(G)|+ 1)− (γ(x) + γ(y) + γ(xy))
= 3(|V (G)|+ |E(G)|+ 1)− k,
32
that is, 3(|V (G)|+ |E(G)|+ 1)− k is the magic constant corresponding to γ′.
From the fact that the dual of an edge-magic labeling of a graph G is also an
edge-magic labeling of G, the following theorem is immediately true.
Theorem 2.1.8. For a connected bipartite graph G = (X, Y ), exactly one of the
following is true:
(i) G does not have a b-edge consecutive magic labeling for any b;
(ii) G has a b-edge consecutive magic labeling for b ∈ {0, |X|+ |Y |};
(iii) G is a tree having a b-edge consecutive magic labeling for each b = 0, |X|,
|Y |, |X|+ |Y |.
Proof. We suppose thatG has a b-edge consecutive magic labeling λ and denote the
dual of λ by λ′. By Theorem 2.1.2, b = 0, |X|, |Y |, or |X|+ |Y |. Since |V (G)| =
|X| + |Y |, it is true that λ(X ∪ Y ) = {1, . . . , |X|+ |Y |} and λ(E(G)) = {|X| +
|Y |+1, . . . , |X|+ |Y |+ |E(G)|} if and only if λ′(X∪Y ) = {|E(G)|+1, . . . , |X|+
|Y |+ |E(G)|} and λ′(E(G)) = {1, . . . , |E(G)|}. Similarly, λ(X) = {1, . . . , |X|},
λ(E(G)) = {|X|+1, . . . , |X|+|E(G)|}, and λ(Y ) = {|X|+1+|E(G)|, . . . , |X|+
|Y |+E(G)} if and only if λ′(Y ) = {1, . . . , |Y |}, λ′(E(G)) = {|Y |+1, . . . , |Y |+
|E(G)|}, and λ′(X) = {|Y | + 1 + |E(G)|, . . . , |X| + |Y | + |E(G)|}. Thus, if
b = |X|+ |Y | or |X|, then the dual of a b-edge consecutive magic labeling of G is
a (|V (G)| − b)-edge consecutive magic labeling of G and vice versa. Since λ is the
dual of λ′, this statement is true even for b = 0 or |Y |. Therefore, if b = |X| or |Y |,
then, from the observation by Marr and Wallis [58] together with Theorem 2.1.7,
the statement (iii) follows. If neither (i) nor (iii) is true, then b = 0 or |X|+ |Y |. By
the above argument again, the statement (ii) is immediately true.
33
Given a b-edge consecutive magic labeling λ of a connected bipartite graph,
there is a way of deducing a b-edge consecutive magic labeling with a new magic
constant from λ without using the dual of edge magic labeling.
Proposition 2.1.9. If λ is a b-edge consecutive magic labeling of a connected bi-
partite graph G = (X, Y ) with magic constant k, then the mapping λ∗ : V (G) ∪
E(G) → {1, . . . , |V (G)|+ |E(G)|} defined by
(i) λ∗(x) = |V (G)|+2|E(G)|+1−λ(x) for a vertex x and λ∗(xy) = |E(G)|+
1−λ(xy) for two adjacent vertices x and y is also a b-edge consecutive magic
labeling of G with magic constant 2|V (G)|+ 5|E(G)|+ 3− k if b = 0;
(ii) λ∗(x) = |X|+|Y |+1−λ(x) for a vertex x and λ∗(xy) = 2|V (G)|+|E(G)|+
1−λ(xy) for two adjacent vertices x and y is also a b-edge consecutive magic
labeling of G with magic constant 4|V (G)|+ |E(G)|+ 3− k if b = |V (G)|;
(iii) λ∗(x) = |X|+1−λ(x) for a vertex x ∈ X , λ∗(y) = 2|X|+ |Y |+2|E(G)|+
1 − λ(y) for a vertex y ∈ Y , and λ∗(xy) = 2|X| + |E(G)| + 1 − λ(xy) for
two adjacent vertices x and y is also a b-edge consecutive magic labeling of
G with magic constant 5|X|+ |Y |+ 3|E(G)|+ 3− k if b = |X|;
(iv) λ∗(x) = |X| + 2|Y | + 2|E(G)| + 1 − λ(x) for a vertex x ∈ X , λ∗(y) =
|Y |+1− λ(y) for a vertex y ∈ Y , and λ∗(xy) = 2|Y |+ |E(G)|+1− λ(xy)
for two adjacent vertices x and y is also a b-edge consecutive magic labeling
of G with magic constant |X|+ 5|Y |+ 3|E(G)|+ 3− k if b = |Y |.
Proof. Suppose that b = |X|. Then λ(X) = {1, . . . , |X|}, λ(Y ) = {|X|+|E(G)|+
1, . . . , |X| + |Y | + |E(G)|} and λ(E(G)) = {|X| + 1, . . . , |X| + |E(G)|}. By
34
definition, it is easy to check that λ∗ is also an |X|-edge consecutive magic labeling
of G. Furthermore,
= (|X|+ 1− λ(x)) + (2|X|+ |Y |+ 2|E(G)|+ 1− λ(y))
+ (2|X|+ |E(G)|+ 1− λ(xy))
= 5|X|+ |Y |+ 3|E(G)|+ 3− k,
which is a constant number. For the remaining cases, it can be similarly checked.
2.1.2 Consecutive edge magic labelings for trees
In the previous section, we have shown that if a connected bipartite graph G has
a b-edge consecutive magic labeling for some b ∈ {1, . . . , |V (G)| − 1}, then G
is a tree. In this section, we study consecutive edge magic labelings of interesting
families of trees.
As mentioned in Remark 2.1.3, the double star Sm,n has an (m + 1)-edge con-
secutive magic labeling and an (n + 1)-edge consecutive magic labeling. We show
that there are only two such labelings for each of m, n.
Proposition 2.1.10. For some positive integers m and n, the double star Sm,n has
only two (m+1)-edge consecutive magic labelings (resp. (n+1)-edge consecutive
magic labelings) both of which have magic constant 4m+2n+6 (resp. 4n+2m+6).
Proof. We may regard G := Sm,n as a bipartite graph with bipartition (X, Y ) with
|X| = m + 1 and |Y | = n + 1. Let λ be an (m + 1)-edge consecutive magic
35
labeling of G and k be the magic constant of λ. Then, by the definition of (m+ 1)-
edge consecutive magic labeling and Proposition 2.1.1,
λ(X) = [m+ 1] and λ(Y ) = {2m+ n+ 3, . . . , 2m+ 2n + 3} (2.7)
or |X| = |Y | and λ(Y ) = [m+ 1]. By symmetry, we may assume (2.7).
Let u and v be the central vertices of G and let λ(u) = α and λ(v) = β.
Without loss of generality, we may assume that u ∈ X and v ∈ Y . As u and v
belong to different partite sets and the vertices other than the central vertices form
an independent set, we know from (2.7) that the labeling λ is completely determined
by α and β. Therefore it suffices to show that there are only two possible pairs of
integers for (α, β). By (2.7) and the assumption that u ∈ X and v ∈ Y ,
A := {λ(u) + λ(y) | y ∈ Y } = {α + 2m+ n+ 3, · · · , α + 2m+ 2n+ 3}
and
B := {λ(v) + λ(x) | x ∈ X} = {β + 1, · · · , β +m+ 1}.
Suppose γ ∈ A∩B. Then γ = λ(u)+ λ(y∗) = λ(v)+ λ(x∗) for some x∗ ∈ X and
y∗ ∈ Y . By the definition of λ, k − λ(uy∗) = k − λ(vx∗) for the magic constant k
of λ, which implies λ(uy∗) = λ(vx∗). Thus we reach a contradiction unless x∗ = u
and y∗ = v. Thus A∩B = {λ(u)+λ(v)} and so |A∩B| = 1. Moreover, X ∪Y =
V (G) and each edge is incident to u or v, so A∪B = {λ(x) + λ(y) | xy ∈ E(G)}.
Since λ is a consecutive edge magic labeling, {k− λ(xy) | xy ∈ E(G)} is a set
of m+n+1 consecutive integers and therefore {λ(x)+λ(y) | xy ∈ E(G)} is a set
ofm+n+1 consecutive integers. SinceA∪B = {λ(x)+λ(y) | xy ∈ E(G)},A∪B
is a set of m+ n + 1 consecutive integers. This together with the fact |A ∩B| = 1
imply that there are only two possible cases:
α + 2m+ n+ 3 = β +m+ 1 or α + 2m+ 2n+ 3 = β + 1.
36
Assume the former. Since 2m+ n + 3 ≤ β and α ≤ m+ 1,
(2m+ n+ 3) +m+1 ≤ β +m+ 1 = α+ 2m+ n+ 3 ≤ (m+ 1) + 2m+ n+ 3.
Since the left hand side of the first inequality and the right hand side of the second
inequality both equal 3m+n+4, we have β+m+1 = 3m+n+4 and α+2m+
n + 3 = 3m + n + 4. Hence α = m + 1 and β = 2m + n + 3. Now assume the
latter. Since β ≤ 2m+ 2n+ 3 and 1 ≤ α,
1 + 2m+ 2n + 3 ≤ α + 2m+ 2n+ 3 = β + 1 ≤ 2m+ 2n+ 3 + 1.
Thus β = 2m+ 2n + 3 and α = 1. We can easily check that the magic constant is
4m+ 2n+ 6 in both cases.
By symmetry, the double star Sm,n has only two (n+1)-edge consecutive magic
labelings and their magic constant is 2m+ 4n+ 6.
Proposition 2.1.10 tells us that magic constants of (m + 1)-edge consecutive
magic labelings and (n+ 1)-edge consecutive magic labelings for a double star are
unique. As a matter of fact, the magic constants of double stars are of specific form.
Theorem 2.1.11. The magic constants of consecutive edge magic labelings of the
double star Sm,n are in the form of dt+6 for some nonnegative integer t where d is
the greatest common divisor of m and n.
Proof. Suppose that the double star G := Sm,n has a b-edge consecutive magic
labeling λ and k is the magic constant of λ. Let x and y be the central vertices with
degrees m + 1 and n + 1, respectively, of G and λ(x) = i and λ(y) = j. Then
37
k(m+ n+ 1) = ∑
2 +mλ(x) + nλ(y)
= (m+ n + 1)(2m+ 2n+ 5) +mi+ nj + 1. (2.8)
Since k is a positive integer, mi+ nj +1 is a multiple of m+ n+1 by the equality
(2.8). That is, mi + nj + 1 = l(m + n + 1) for some positive integer l. Then, by
(2.8),
k = 2m+ 2n+ 5 + l (2.9)
Since it is impossible for both i and j to equal 1, we have l ≥ 2. Let d be the greatest
common divisor of m and n. Then m = dm′ and n = dn′ for relatively prime
positive integersm′ and n′. Suppose that d = 1, that is,m and n are relatively prime.
Then, since l ≥ 2, by the Bezout’s identity, l−1 = µ1m+ν1n or l = µ1m+ν1n+1
for some integers µ1 and ν1. Then, by (2.9), k = 2m + 2n + µ1m + ν1n + 6 =
d(2m+ 2n+ µ1m+ ν1n) + 6.
Now suppose d ≥ 2. By the division algorithm, l = dq + r for some integers q
and r with 0 ≤ r ≤ d− 1. Then
mi+ nj + 1 = (m+ n + 1)(dq + r)
or
m(i− dq − r) + (j − dq − r)n− dq = r − 1.
Since the left hand side is divisible by d, r − 1 is a multiple of d. Since r ≤ d− 1,
r = 1 and so l = dq + 1. Hence, by (2.9),
k = 2m+ 2n + 5 + (dq + 1) = d(2m′ + 2n′ + q) + 6
38
and we complete the proof.
We may regard Sm,n as a bipartite graph with bipartition (X, Y ) with |X| =
n + 1 and |Y | = m + 1. From the (n + 1)-edge consecutive magic labeling given
in [56], we know that 2(n + 1) + 4(m + 1) = 4m + 2n + 6 is a magic constant
for n+ 1. For the β-edge consecutive magic labeling given in [56] where β = |Y |,
we define : V (Sn1,n2,...,nr ) ∪ E(Sn1,n2,...,nr
) → {1, 2, . . . , 2 ∑r
i=1 ni + 2r − 1}
by (y) = λ(y) for each y ∈ Y ; (x) = λ(x) − α − β + 1 for each x ∈ X;
(xy) = λ(xy) + α for each pair of adjacent vertices x and y of Sn1,n2,...,nr . It
can easily be checked that is a super edge-magic labeling by the fact that λ is a
β-edge consecutive magic labeling. Now we take two adjacent vertices x and y of
Sm,n. Recall that (X, Y ) is its bipartition. Then we may assume that x ∈ X and
y ∈ Y . By the definition,
(x) + (y) + (xy) = (λ(x)− α− β + 1) + λ(y) + (λ(xy) + α)
= 2α+ 3β + 1 = 3m+ 2n+ 6.
Then, by Proposition 2.1.9(ii),
= 4(m+ n+ 2) + (m+ n+ 1) + 3− (3m+ 2n+ 6)
= 2m+ 3n+ 6
is another magic constant of G for b = m + n + 2. Furthermore, recalling that
the dual γ′ of a b-edge consecutive magic labeling γ with a magic constant k is a
(|V (G)| − b)-edge consecutive magic labeling with the magic constant 3(|V (G)|+
|E(G)|+ 1)− k, we obtain
3(|V (G)|+ |E(G)|+ 1)− (2m+ 3n+ 6) = 4m+ 3n+ 6
39
is a magic constant of G for b = 0. Then, by Proposition 2.1.9(i),
2|V (G)|+ 5|E(G)|+ 3− (4m+ 3n+ 6)
= 2(m+ n+ 2) + 5(m+ n+ 1) + 3− (4m+ 3n+ 6)
= 3m+ 4n+ 6
is another magic constant for b = 0. By the symmetry, 4(n + 1) + 2(m + 1) =
2m+ 4n+ 6 is a magic constant for b = m+ 1.
In the rest of paper, we take a look at a special type of a lobster which is obtained
from a star graph G by attaching a leaf to each leaf of G. For a positive integer p,
we denote by Lp the lobster obtained from a star graph with p leaves in such a way.
In addition, we denote the center of Lp by x, the vertices at distance 1 from x by y1,
. . ., yp, the vertex adjacent to yi by xi for each i = 1, . . ., p. Now the following is
true for Lp.
Theorem 2.1.12. For p ≥ 3, Lp has a b-edge consecutive magic labeling if and
only if b ∈ {0, 2p+ 1}.
Proof. Kim and Park [38] showed that Lp has a (2p + 1)-edge consecutive magic
labeling. Thus, by Theorem 2.1.8, the ‘if’ part is true.
Now we show the ‘only if’ part. By Theorem 2.1.2, b ∈ {0, p, p+1, 2p+1}. For
notational convenience, we denote {1, . . . , p+1} by [p+1]. To reach a contradiction,
suppose that b = p + 1. Then there is a (p + 1)-edge consecutive magic labeling λ
of Lp such that
By Proposition 2.1.1,
λ({x, x1, . . . , xp}) = [p+ 1] and λ({y1, . . . , yp}) = {3p+ 2, . . . , 4p+ 1}.
40
Set λ(x) = i. Then i ∈ [p + 1]. Let k be the magic constant corresponding to λ.
Then the values of λ for the edges joining x and the vertices y1, . . ., yp are
k − 4p− i− 1, . . . , k − 3p− i− 2.
Suppose that i+j ∈ [p+1] for some integer j. Then i+j is assigned to a vertex
in {x, x1, . . . , xp}. The set of possible values of λ for the edge joining a vertex in
{y1, . . . , yp} and the vertex labeled with i+ j is
A(i, j) := {k − [(i+ j) + (4p+ 1)], . . . , k − [(i+ j) + (3p+ 2)]}.
Then
A(i, 1) = {k − 4p− i− 2, . . . , k − 3p− i− 3}
and
A(i,−1) := {k − 4p− i, . . . , k − 3p− i− 1}.
We first note that all the elements of A(i, 1) except k− 4p− i− 2 = k− [(i+1)+
(4p+1)] have been already assigned to the edges joining x and y1, . . ., yp, and so the
vertex labeled with 4p+ 1 must be joined with the one labeled with i+ 1. We also
note that all the elements ofA(i,−1) except k−3p− i−1 = k− [(i−1)+(3p+2)]
are occupied by the edges joining x and y1, . . ., yp, and so the vertex labeled with
3p+ 2 must be joined with the one labeled with i− 1
Now suppose that i ≥ 3. Then i − 2 ∈ [p + 1] and k − [(i − 2) + (3p + 2)] =
k − 3p− i is the only element in A(i,−2) that was not assigned to edges joining x
and y1, . . ., yp. Thus the vertex labeled with i − 2 and the one labeled with 3p + 2
should be joined. However, i − 1 ∈ [p + 1], and the vertex labeled with 3p + 2
must be joined to the one labeled with i− 1 by the above argument and we reach a
contradiction.
41
Now suppose that i = 2. Then i+ 2 = 4 ∈ [p + 1] since p ≥ 3 and
A(i, 2) = {k − [(i+ 2) + (4p+ 1)], . . . , k − [(i+ 2) + (3p+ 2)]},
in which k−[(i+2)+(4p+1)] = k−4p−i−3 is the only available label. However,
in that case, the vertex labeled with i+ 2 and the one labeled with 4p+1 should be
joined, which is a contradiction as the vertex labeled with 4p + 1 is already joined
to the one labeled with i+ 1.
Now suppose that i = 1. Then the values of λ for the edges joining x and
vertices y1, . . ., yp are k − 4p− 2, . . ., k − 3p− 3. Moreover, 2 = i+ 1 ∈ [p + 1]
since p ≥ 3, and
A(1, 1) = {k − 4p− 3, . . . , k − 3p− 4},
in which k− 4p− 3 = k− [(1+ 1)+ (4p+1)] is the only available label. Thus the
vertex labeled with 2 and the one labeled with 4p + 1 are joined. Now 3 = i+ 2 ∈
[p+ 1] since p ≥ 3, and
A(1, 2) = {k − 4p− 4, . . . , k − 3p− 5}
in which k − 4p − 4 = k − [(1 + 2) + (4p + 1)] is the only available label for the
edge incident to the vertex labeled with 3. Then, however, the other end vertex of
the edge must be labeled with 4p+1, which is impossible as the vertex labeled with
4p+ 1 is adjacent to the one labeled with 2.
Suppose that i = p+1. Then the values of λ for the edges joining x and vertices
y1, . . ., yp are k − 5p− 2, . . ., k − 4p− 3. Now for p = i− 1,
A(p + 1,−1) = {k − 5p− 1, . . . , k − 4p− 2}
in which k − 4p− 2 = k − [(p+ 1) + (−1) + (3p+ 2)] is the only available label
for the edge incident to the vertex labeled with p. Thus the vertex labeled p and the
42
9
Figure 2.1: A 2-edge consecutive magic labeling for L1 and a 3-edge consecutive
magic labeling for L2.
one labeled with 3p+ 2 are joined. On the other hand, for p− 1 = i− 2,
A(p+ 1,−2) = {k − 5p, . . . , k − 4p− 1}
in which k − 5p = k − [(p + 1) + (−2) + (4p + 1)] is the only available label for
the edge incident to the vertex labeled with p−1. Then, however, the vertex labeled
with p− 1 and the one labeled with 3p+2 should be joined, which is impossible as
the vertex labeled with 3p+ 2 also must be adjacent to the one labeled with p.
Thus there is no (p + 1)-edge consecutive magic labeling for Lp. Then, by
Theorem 2.1.8, the statement is true.
Remark 2.1.13. For p = 1 or 2, the above theorem is false. We may assign (p+1)-
edge consecutive magic labelings to L1 and L2, which are the paths P3 and P5 of
lengths 2 and 4, respectively (see Figure 2.1).
Remark 2.1.14. By Theorem 2.1.12, we know that the converse of Theorem 2.1.7
is false as the lobster L4 is super edge-magic and graceful, but it does not have a
b-edge consecutive magic labeling for b = 4 or 5. A graceful labeling for L4 is
shown in Figure 2.2.
2.2 Super edge-magic labeling graphs
Since Enomoto et al. [17] introduced the notion of super edge-magic labeling, there
have been a lot of studies finding a super edge-magic labeling for given a graph.
As a part of those researches, it is widely studied to obtain a new super edge-magic
labeling from specific families of super edge-magic graphs. In this chapter, we pro-
vided a way of obtaining super edge-magic graphs from super edge-magic graphs
via graph operations.
We first present a useful lemma provided by Enomoto et al. [17].
Lemma 2.2.1. ( [17]) If G is a super edge-magic graph, then |E(G)| ≤ 2|V (G)| −
3.
Let G be a super edge-magic graph and f be a super edge-magic labeling of
G with magic constant k. Suppose that f(xy) < f(uv) for some xy, uv ∈ E(G).
Then k − f(xy) > k − f(uv) or f(x) + f(y) > f(u) + f(v). Since f is super
edge-magic labeling, there are edges x1y1 and x2y2 such that f(x1y1) = |V (G)|+1
44
and f(x2y2) = |V (G)| + |E(G)|. Then f(x1) + f(y1) = minxy∈E(G) f(x) + f(y)
and f(x2) + f(y2) = maxxy∈E(G) f(x) + f(y). Since f(V (G)) = {1, . . . , |V (G)},
1 + 2 ≤ f(x) + f(y) ≤ |V (G)| − 1 + |V (G)| for xy ∈ V (G). Therefore we obtain
the following inequalities:
1 + 2 + (|V (G)|+ |E(G)|) ≤ k ≤ |V (G)|+ (|V (G)| − 1) + (|V (G)|+ 1).
Thus a magic constant k of a super edge-magic graph G satisfies that
|V (G)|+ |E(G)|+ 3 ≤ k ≤ 3|V (G)|. (2.10)
The following lemma shows that it is sufficient to consider only vertex-labelings
of a graph to determine whether it is super edge-magic or not.
Lemma 2.2.2. ( [18]) A graph G is super edge-magic if and only if there exists a
bijective function f : V (G) → {1, . . . , |V (G)|} such that the set
S = {f(u) + f(v) | uv ∈ E(G)}
consists of |E(G)| consecutive integers. In such a case, f extends to a super edge-
magic labeling of G with magic constant k = |V (G)| + |E(G)| + s, where s =
min(S) and
S = {k − (|V (G)|+ 1), k − (|V (G)|+ 2), . . . , k − (|V (G)|+ |E(G)|)}.
2.2.1 Obtaining a new super edge-magic graph through graph
operations
One of the interesting topics on super edge-magic labeling is extending a super
edge-magic labeling of a graph to a super edge-magic labeling of a bigger graph
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which contains the original graph. In this subsection, we introduce a way of obtain-
ing a new super edge-magic graph from two super edge-magic graphs.
We first present the following lemma derived from Lemma 2.2.2.
Lemma 2.2.3. Given graph G, suppose that there exists a bijective function f from
V (G) to {1, . . . , |V (G)|} such that {f(u) + f(v) | uv ∈ E(G)} is partitioned into
two subsets L1 and L2 consisting of consecutive integers satisfying max(L1) + 1 <
min(L2). If there exists an edge setB ofG such that |B| = min(L2)−max(L1)−1,
and {f(u) + f(v) | uv ∈ B} = {max(L1) + 1, . . . ,min(L2)− 1}, then f extends
to a super-edge magic labeling of the new graph obtained by adding the edges in B
to G.
Proof. Since {f(u) + f(v) | uv ∈ E(G)} ∪ {f(u) + f(v) | uv ∈ B} is a set of
consecutive integers, the lemma immediately comes from Lemma 2.2.2.
Let f and g be super edge-magic labelings of graphs G and H , respectively. We
label the vertices ofG as v1, . . . , v|V (G)| and the vertices ofH asw1, . . . , w|V (H)|. We
define a function h : V (G)∪V (H) → {1, . . . , |V (G)|+ |V (H)|} by h(vi) = f(vi)
and h(wi) = g(wi)+ |V (G)| for all i. We call h the concatenation of f and g. Then
h is bijective,
and
{h(wi) | 1 ≤ i ≤ |V (H)|} = {|V (G)|+ 1, . . . , |V (G)|+ |V (H)|}. (2.11)
By using the notion of the concatenation, we present a way of obtaining a new
super edge-magic graph from two super edge-magic graphs satisfying certain con-
dition.
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Theorem 2.2.4. Let G and H be super edge-magic graphs. If |V (G)|−1 ≤ |E(G)|
and |V (H)| − 1 ≤ |E(H)|, then there exists a super edge-magic graph T such that
(G ∪ H) ⊂ T ⊂ (G ∨ H) and T contains at least one edge joining a vertex in G
and a vertex in H .
Proof. Let f and g be super edge-magic labelings of G and H with magic con-
stant k1 and k2, respectively. Let |V (G)| = n1, |E(G)| = m1, |V (H)| = n2
and |E(H)| = m2. We label the vertices V (G) = {v1, . . . , vn1 } and V (H) =
{w1, . . . , wn2 } so that f(vi) = i and g(wj) = j for each i and j, 1 ≤ i ≤ n1,
1 ≤ j ≤ n2. Let h be the concatenation of f and g.
Since f and g are super edge-magic labelings, by Lemma 2.2.2,
L1 : = {h(vi) + h(vj) | vivj ∈ E(G)}
= {f(vi) + f(vj) | vivj ∈ E(G)}
and
= {k2 − n2 −m2 + 2n1, . . . , k2 + n2 − 1 + 2n1}.
By (2.10), ni+mi+3 ≤ ki ≤ 3ni for i = 1, 2. Therefore k1−k2 ≤ 3n1−n2−m2−3.
Thus
k1 − n1 − 1 ≤ k2 − n2 −m2 + 2n1 − 4 < k2 − n2 −m2 + 2n1 − 1.
Hence the largest element of L1 is strictly less than the smallest element of L2. For
each i = 1, . . . , ni, we define sets
Ai = {h(vi) + h(wj) | 1 ≤ j ≤ n2}.
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By (2.11), Ai = {h(vi) + n1 + 1, . . . , h(vi) + n1 + n2}, so
∪n1
i=1Ai = {n1 + 2, . . . , 2n1 + n2}.
Since n1+m1+3 ≤ k1 and n1 ≤ m1+1, we have 2n1+2 ≤ k1 or n1+2 ≤ k1−n1.
In addition, since k2 ≤ 3n2 and n2 ≤ m2 + 1, we have k2 ≤ 2n2 + m2 + 1 or
k2 −m2 − n2 + 2n1 − 1 ≤ 2n1 + n2. Therefore
{k1 − n1, . . . , k2 − n2 −m2 + 2n1 − 1} ⊂ ∪n1
i=1Ai.
Thus for each element t in {k1 − n1, . . . , k2 − n2 −m2 + 2n1 − 1}, there is (it, jt)
such that h(vit) + h(wjt) = t. Then we join vertex vit and wjt by an edge for each
t ∈ {k1 − n1, . . . , k2 − n2 −m2 + 2n1 − 1}. Let
B = {vitwjt | t = k1 − n1, . . . , k2 − n2 −m2 + 2n1 − 1}.
Then B clearly satisfies two properties of Lemma 2.2.3. In addition, the difference
of the largest element of L1 and the smallest element of L2 is at least two and so
B 6= ∅. Therefore there is an edge in T which joining a vertex of G and a vertex of
H .
The following corollary immediately follows from Theorem 2.2.4.
Corollary 2.2.5. Let G be a super edge-magic graph such that |V (G)| − 1 ≤
|E(G)|. Then there exists a super edge-magic graph T such that (G ∪ G) ⊂ T ⊂
(G ∨G).
Corollary 2.2.6. Let G and H be super edge-magic graphs. Then there exists a
connected super edge-magic graph T such that (G ∪H) ⊂ T ⊂ (G ∨H).
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11
2
2
3
3
4
G H
Figure 2.3: Super-edge magic labelings of a caterpillar G and 5-cycle H , respec-
tively.
14 T
Figure 2.4: Super-edge magic labeling a graph T such that (G∪H) ⊂ T ⊂ (G∨H).
Proof. A connected graph G satisfies the inequality |V (G)| − 1 ≤ |E(G)|. There-
fore, by Theorem 2.2.4, there exists a super edge-magic graph T such that (G ∪
H) ⊂ T ⊂ (G ∨ H) and T contains an edge joining a vertex of G and a vertex of
H . Since G and H are connected, T is also connected.
We have shown that caterpillars are super edge-magic and a cycle of odd length
is super edge-magic. See Figure 2.3 for an illustration. Since the caterpillar G and
the 5-cycleH in Figure 2.3 are connected graphs, there is a super edge-magic graph
T such that (G ∪G) ⊂ T ⊂ (G ∨G) by Corollary 2.2.6 as shown in Figure 2.3.
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A 2-regular graph is disjoint union of cycles. We show that a 2-regular super
edge-magic graph can be extended to a 3-regular super edge-magic graph.
Proposition 2.2.7. Let G be a 2-regular super edge-magic graph. Then there is a
3-regular super edge-magic graph T such that (G ∪G) ⊂ T ⊂ (G ∨G).
Proof. Since G is a 2-regular graph, G is a union of disjoint cycles. Therefore the
number of vertices of G and the number of edges of G are the same. We denote by
n the number of vertices of G. Let f be super edge-magic labeling of G and k be
the magic constant of f . Then
k · |E(G)| = ∑
2 .
Therefore k = 2n + 1 + n+1 2
and G has an odd number of vertices. Since k is an
integer, n = 2t + 1 for some nonnegative integer t and so G has an odd number of
vertices.
Let T = G ∪ G and h be the concatenation of f and f . Since k = 5t + 4 and
|V (G)| = |E(G)| = 2t + 1, by Lemma 2.2.3,
{h(v)+ h(w) | vw ∈ E(T )} = {t+2, . . . , 3t+2}∪ {5t+4, . . . , 7t+4}. (2.12)
We take the edge joining the vertex in one of the two copies ofG in T with label
i and the vertex in the other of the two copies of G in T with label 4t + 3 − 2i for
each i, 1 ≤ i ≤ t in T and denote the set of these edges by E1. Then we take the
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edge joining the vertex in one of the two copies of G in T with label t + i and the
vertex in the other of the two copies of G in T with label 4t + 4 − 2i for each i,
1 ≤ i ≤ t in T and denote the set of these edges by E2.
Since h is bijective, E1 and E2 are well-defined. Each edge in E1 is incident to
a vertex with label i and a