local antimagic coloring of graphs - prof....
TRANSCRIPT
Local Antimagic Coloring of Graphs
Prof. Slamin
Universitas Jember
Workshop KK Kombinatorika: Pewarnaan dan Pelabelan pada Graf
Jurusan Matematika Universitas Andalas
Padang, 25 April 2019
Overview
1. Local Antimagic Vertex Coloring
2. Local Antimagic Total Vertex Coloring
3. Super Local Antimagic Total Vertex Coloring
4. Local Antimagic Edge Coloring
5. Conclusion
2
Introduction
Definition [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
The local antimagic labeling on a graph G with |V | vertices and |E |edges is defined to be an assignment f : E → {1, 2, · · · , |E |} so that
the weights of any two adjacent vertices u and v are distinct, that is,
w(u) 6= w(v) where w(u) = Σe∈E(u)f (e) and E (u) is the set of edges
incident to u.
• Any local antimagic labeling induces a proper vertex coloring of G
where the vertex u is assigned the color w(u).
• The local antimagic chromatic number, denoted by χla(G ), is the
minimum number of colors taken over all colorings induced by local
antimagic labelings of G .
• For any graph G , χla(G ) ≥ χ(G ) and the difference χla(G )− χ(G )
can be arbitrarily large.
4
Cycle
Theorem [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
For the cycle Cn on n ≥ 3 vertices, χla(Cn) = 3
Proof.
• Label the edges of Cn using the following formula.
f (vivi+1) =
{n − i−1
2 if i is oddi2 if i is even
• The weight of vertices are
w(vi ) =
n if i is odd, i 6= 1
n + 1 if i is even
2n − b n2c if i = 1
• Thus χla(Cn) ≤ 3.
6
Cycle (cont.)
Proof (cont.)
• Suppose n is even and there exists f that induces a 2-coloring of Cn.
• Let x be the color of vi if i is odd and y if i is even.
• Then n2 (x + y) = 2[ n(n+1)
2 ] and hence x + y = 2n + 2.
• If f (vivi+1) = n, then w(vi ) and w(vi+1) are at least n+ 1 and n+ 2.
• Thus x + y ≥ 2n + 3, which is a contradiction.
• So χla(Cn) ≥ 3 if n is even.
• Further, χ(Cn) = 3 if n is odd and hence χla(Cn) ≥ 3.
• Therefore χla(Cn) = 3.
7
Tree
Theorem [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
For any tree T with l leaves, χla(T ) ≥ l + 1.
Figure 2: Tree with l leaves
8
Path
Theorem [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
For path Pn on n ≥ 3 vertices, χla(Pn) = 3.
Figure 3: The local antimagic vertex coloring of P7 with χla(P7) = 3
9
Friendship graph
Theorem [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
For friendship graph Fn with n ≥ 2, χla(Fn) = 3.
Theorem [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
For friendship graph Fn with n ≥ 2 by removing an edge e, χla(Fn −{e}) = 3.
Figure 4: The local antimagic vertex coloring of F4 and F4 − {e}10
Complete bipartite graph
Theorem [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
For complete bipartite graph Km,n with m, n ≥ 2, χla(Km,n) = 2 if and
only if m ≡ n (mod 2).
Theorem [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
For complete bipartite graph K2,n with n ≥ 2, χla(K2,n) = 2 for even
n ≥ 2 and χla(K2,n) = 3 for odd n ≥ 3 or n = 2.
11
Graph Ln
Theorem [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
Graph Ln for n ≥ 2 that is obtained by inserting a vertex to each edge
vvi , 1 ≤ i ≤ n − 1, of the star, χla(Ln) = n + 1.
Figure 5: The local antimagic vertex coloring of L7 with χla(L7) = 8
12
Wheel
Theorem [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
Wheel Wn of order n + 1 for n ≥ 3, χla(Wn) = 4 if n ≡ 1, 3 (mod 4),
χla(Wn) = 3 if n ≡ 2 (mod 4), and 3 ≤ χla(Wn) ≤ 5 if n ≡ 0 (mod 4).
Figure 6: The local antimagic vertex coloring of W6 with χla(W6) = 3
13
Fan
Corollary
Fan fn of order n + 1 for n ≥ 6 and n ≡ 2 (mod 4), χla(fn) = 3.
Figure 7: The local antimagic vertex coloring of f6 with χla(f6) = 3
14
Graph resulting from operation
Theorem [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
For the graph H = G + K2 where G is a graph of order n ≥ 4, then
χla(G ) + 1 ≤ χla(H) ≤
{χla(G ) + 1 for n is even
χla(G ) + 2 otherwise.
15
Kite
Theorem [Nazula, S, Dafik (2018)]
For the kite Ktn,m with n ≥ 3 and m ≥ 1, χla(Ktn,m) = 3.
Proof.
• Label the edges of Ktn,m using the following formula.
f (uiui+1) =
{m+i2 if m + i is even
m+3−i2 + b n2c+ b n−12 c if m + i is odd
f (unu1) =
{b n2c+ m+1
2 if m is odd
b n+12 c+ m
2 if m is even
f (u1v1) =
{m2 if m is evenm+12 + n if m is odd
f (vjvj+1) =
{m−j2 if m + j is even
m+j+12 + n if m + j is odd
16
Kite (cont.)
Proof (cont.)
• The weight of vertices are
w(ui ) =
3m2 + 2b n+1
2 c+ b n2c if m is even and i = 1
n + m + 1 if m + i is even
n + m if m + i is odd3m+3
2 + b n2c+ n if m is odd and i = 1
w(vi ) =
{n + m if m + j is even
n + m + 1 if m + j is odd
• Thus χla(Ktn,m) ≤ 3.
• The lower bound uses χla(Cn) due to Arumugam et al.
• Since χla(Cn) = 3 and the kite Ktn,m contains contains a subgraph
that is isomorphic to the Cn, then χla(Ktn,m) ≥ 3.
• Therefore χla(Ktn,m) = 3.
17
Cycle with two neighbour pendants
Theorem [Nazula, S, Dafik (2018)]
For cycle with two neighbour pendants Cpn with n ≥ 3, χla(Cpn) = 4.
Proof.
• Label the edges of Cpn using the following formula.
f (uiui+1) =
{i+12 if i is odd
n + 1− i2 if i is even
f (unu1) = dn + 1
2e
f (uivi ) = n + i if i = 1, 2
• The weight of vertices are
w(ui ) =
b 3n+6
2 c if i = 1
2n + 3 if i = 2
n + 2 if i is odd and i ≥ 3
n + 1 if i is even and i ≥ 4
w(vi ) = n + i if i = 1, 2 19
Cycle with two neighbour pendants (cont.)
Proof (cont.)
• Thus χla(Cpn) ≤ 4.
• To show the lower bound, we suppose that f (u1v1) = m1 and
f (u2v2) = m2.
• Then w(v1) = m1, w(v2) = m2, w(u1) > m1 and w(u2) > m2
• Clearly, w(v1) 6= w(v2).
• As u1 is neighbour of u2, then w(u1) 6= w(u2).
• This implies that χla(Cpn) ≥ 4.
• We conclude that χla(Cpn) = 4.
20
Cycle with two neighbour pendants (cont.)
Figure 9: The local antimagic vertex coloring of Cp6 with χla(Cp6) = 4
21
Corona Product of Graphs
Theorems [Arumugam, Lee, Premalatha, Wang (2018+)]
• χla(Pn � K1) = n + 2 for n ≥ 4.
• χla(Pn � Km) = mn + 2 for m ≥ 2 and n ≥ 2.
• χla(Cn � K1) = n + 2 for n ≥ 4.
• χla(Cn � K2) = 2n + 2 for n ≥ 4.
• χla(C3 � K2) = 3m + 3 for m ≥ 2.
• χla(Cn � Km) = mn + 3 for odd n ≥ 5 except finitely many m?s.
• χla(Kn � K1) = 2n − 1 for n ≥ 3.
• χla(Kn � Km) = mn + n for m ≥ 2 and n ≥ 3.
22
Local Antimagic Total Vertex Coloring
Definition [Putri, Dafik, Agustin, Alfarisi (2018)]
The local vertex antimagic total labeling on a graph G with |V | vertices
and |E | edges is defined to be an assignment f : V∪E → {1, 2, · · · , |V |+|E |} so that the weights of any two adjacent vertices u and v are distinct,
that is, w(u) 6= w(v) where w(u) = f (u) + Σe∈E(u)f (e) and E (u) is the
set of edges incident to u.
• Any local vertex antimagic total labeling induces a proper vertex
coloring of G where the vertex u is assigned the color w(u).
• The local antimagic total chromatic number, denoted by χlat(G ), is
the minimum number of colors taken over all colorings induced by
local vertex antimagic total labelings of G .
• For any graph G , χlat(G ) ≥ χ(G ).
24
Particular class of trees
Putri, Dafik, Agustin, Alfarisi (2018)
• For star Sn with n ≥ 2, χlat(Sn) = 2.
• For double star Sn,m with n ≥ 2 and m ≥ 2, χlat(Sn,m) ≤ 3.
• For banana tree Bm,n with n ≥ 3 and m ≥ 3,
χlat(Bm,n) ≤
4, if n odd, m odd
5, if n odd, m even
6, if n even, m even
26
Wheel
Theorem [S, Dafik, Hasan (2018)]
For wheel Wn of order n + 1,
χlat(Wn) =
{3, n ≡ 0 (mod 2)
4, n ≡ 1 (mod 2).
Proof.
• Label all vertices and edges of Wn using the following formula.
f (xixi+1) = i , for 1 ≤ i ≤ n − 1
f (xnx1) = n
f (cxi ) = 2n + 1− i , for 1 ≤ i ≤ n
f (xi ) =
2n + 2, for i = 1
3n, for i = 2, n odd
3n − i + 2− (−1)i+n, untuk 2 ≤ i ≤ n
f (c) = 3n + 1.27
Wheel (cont.)
Proof (cont.).
• The weight of the vertices are
w(xi ) =
5n + 1, for (i + n) even
5n + 2, for i = 2, n odd
5n + 3, for (i + n) odd or i = 1, n odd
w(c) =n(3n + 7)
2+ 1.
• Thus χlat(Wn) ≤ 3 for n even and χlat(Wn) ≤ 4 for n odd.
• Since χlat(Wn) ≥ χ(Wn) = 3 for n even and χlat(Wn) ≥ χ(Wn) = 4
for n odd.
• Therefore, χlat(Wn) = 3 for n even and χlat(Wn) = 4 for n odd.
28
Other wheel related graphs
Theorems [S, Dafik, Hasan (2018)]
• For fan Fn of order n ≥ 3, χlat(fn) = 3
• For friendship graph fn of order 2n + 1 with n ≥ 2, χlat(fn) = 3
30
Broom
Theorem [Nikmah, S, Hobri (2018)]
For broom Bn,m with m ≥ 3 and n −m ≥ 2, χla(Bn,m) = n −m + 2
Theorem [Nikmah, S, Hobri (2018)]
For double broom Bn,m with m ≥ 3 and n −m ≥ 2,
1. χlat(Bn,m) ≤ 3, and
2. χlat(Bn,m) = 3, if n >7 + 2m +
√48m − 23
2.
31
Prism and Mobius ladder
Theorem [Hasan, S, Dafik (2018)]
For prism Dn of order 2n with n ≥ 3,
χlat(Dn) =
{2, n ≡ 0(mod 2)
3, n ≡ 1(mod 2).
Theorem [Hasan, S, Dafik (2018)]
For Mobius ladder M2n of order 2n with n ≥ 3,
χlat(M2n) =
{3, n ≡ 0(mod 2)
2, n ≡ 1(mod 2).
32
Super Local Antimagic Total Vertex Coloring
Definition
The super local vertex antimagic total labeling on a graph G with |V |vertices and |E | edges is defined as local vertex antimagic total labeling
where the vertices of G receive the smallest labels, that is, 1, 2, ..., |V |.
• Any super local vertex antimagic total labeling induces a proper
vertex coloring of G where the vertex u is assigned the color w(u).
• The super local antimagic total chromatic number, denoted by
χslat(G ), is the minimum number of colors taken over all colorings
induced by super local vertex antimagic total labelings of G .
• For any graph G , χslat(G ) ≥ χlat(G ) ≥ χ(G ).
34
Some results
Theorem [S, Dafik, Hasan (2018+)]
For cycle Cn of order n ≥ 3,
χlas(Cn) =
{3, if n is odd
2, if n is even.
Theorem [S, Dafik, Hasan (2018+)]
For path Pn of order n ≥ 4, 3 ≤ χslat(Pn) ≤ 4,
36
Gear and Generalised Wheel
Theorem [S, Dafik, Hasan (2018+)]
For gear Jn,1 of order 2n + 1 and odd n ≥ 5, χslat(Jn,1) = 3.
Theorem [S, Dafik, Hasan (2018+)]
For generalized wheel W nm of order mn + 1 where m ≥ 1 and n ≥ 3,
χslat(Wnm) =
{3, n even
4, n odd
37
Local Antimagic Edge Coloring
Definition [Agustin, Hasan, Dafik, Alfarisi, Prihandini (2017)]
The local edge antimagic labeling on a graph G with |V | vertices and
|E | edges is defined to be an assignment f : V → {1, 2, · · · , |V |} so
that the weights of any adjacent edges, that is {w(uv) : w(uv) =
f (u) + f (v), uv ∈ E}, are distinct.
• Any local edge antimagic labeling induces a proper edge coloring of
G where the edge uv is assigned the color w(uv).
• The local antimagic edge chromatic number, denoted by χlea(G ), is
the minimum number of colors taken over all colorings induced by
local edge antimagic labelings of G .
• For any graph G , χlae(G ) ≥ χ′(G ).
39
Cycle
Theorem [Agustin, Hasan, Dafik, Alfarisi, Prihandini (2017)]
For the cycle Cn on n ≥ 3 vertices, χlea(Cn) = χla(Cn) = 3
Figure 13: The local antimagic edge coloring of C6 with χlea(C6) = 3
40
Path
Theorem [Agustin, Hasan, Dafik, Alfarisi, Prihandini (2017)]
For the path Pn on n ≥ 2 vertices, χlea(Pn) = 2.
Figure 14: The local antimagic edge coloring of P7 with χlea(P7) = 2
41
Complete Graph
Theorem [Agustin, Hasan, Dafik, Alfarisi, Prihandini (2017)]
For the complete graph Kn on n ≥ 3 vertices, χlea(Kn) = 2n − 3
42
Wheel related graphs [Agustin, Hasan, Dafik, Alfarisi, Prihandini (2017)]
• Wheel Wn for n ≥ 3, χlea(fn) = n + 2
• Fan graph fn for n ≥ 3, χlea(fn) = n + 1
• Gear graph Jn,1 for n ≥ 3, χlea(Jn,1) = n + 2
• Helm graph Hn for n ≥ 3, χlea(Hn) = n + 3
• Flower graph Fln for n ≥ 3, χlea(Fln) = 2n + 1
• Sun flower graph SFln for n ≥ 3, χlea(SFln) = 2n − 1
43
Corona Product of Graphs
Theorem [Agustin, Hasan, Dafik, Alfarisi, Prihandini (2017)]
For corona product of cycle Cn and m copy of K1 where n ≥ 3 and
m ≥ 1, χlea(Cn �mK1) = m + 3,
Theorem [Agustin, Hasan, Dafik, Alfarisi, Prihandini (2017)]
For corona product of any graph G of order n ≥ 3 and m copy of K1
where m ≥ 1, χlea(G �mK1) = χlea(G ) + m,
44
Local Antimagic Total Edge Coloring
Definition [I.H. Agustin, Dafik, M. Hasan, 2017]
The local edge antimagic total labeling on a graph G with |V | vertices
and |E | edges is defined to be an assignment f : V∪E → {1, 2, · · · , |V |+|E |} so that the weights of any adjacent edges, that is {w(uv) : w(uv) =
f (u) + f (uv) + f (v), uv ∈ E}, are distinct.
• Any local edge antimagic total labeling induces a proper edge
coloring of G where the edge uv is assigned the color w(uv).
• The local antimagic total edge chromatic number, denoted by
χleat(G ), is the minimum number of colors taken over all colorings
induced by local edge antimagic total labelings of G .
• For any graph G , χleat(G ) ≥ χ′(G ).
• If the vertices of G receive the smallest labels, that is, 1, 2, ..., |V |,then the such labeling is called super local edge antimagic total
labeling.
• Any super local edge antimagic total labeling also induces a proper
edge coloring of G . 45
Open Problems
Open Problem 1
Determine the (super) local antimagic (total) (edge) chromatic number
of other class of graphs G .
Open Problem 2 [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]
Characterize the class of graphs G for which χla(G ) = χ(G ).
Open Problem 3
Characterize the class of graphs G for which χlat(G ) = χ(G ), χlea(G ) =
χ′(G ), or χleat(G ) = χ′(G ) and χla(G ) = χlat(G ) = χslat(G ) or
χlea(G ) = χleat(G ) = χsleat(G ).
Open Problem 4
How about local irregular labeling vertex or edge coloring?
49