reverse binary graphs

Mathl. Comput. Modelling Vol. 17, No. 11, pp. 49-60, 1993 0895-7177/93 $6.00 + 0.00

Printed in Great Britain. All rights reserved Copyright@ 1993 Pergamon Press Ltd

REVERSE BINARY GRAPHS1

SAJAL K. DAS

Department of Computer Science, University of North Texas

Denton, TX 76203-3886, U.S.A.

NARSINGH DEO

Department of Computer Science, University of Central Florida

Orlando, FL 32816-0001, U.S.A.

SUSHIL PRASAD

Mathematics and Computer Science Department, Georgia State University

Atlanta, GA 30303, U.S.A.

Abstract-We define a new family of labeled, undirected graphs, called reverse binary graphs (RBG), and study their properties. These graphs are incrementally generated and “succinctly repro sented.” Several important parameters of reverse binary graphs, including diameter, node and edge chromatic numbers, domination number, and genus, are obtained easily. Therefore, RBG’s can be used as inputs for testing and experimenting with graph algorithms. Furthermore, like other families of graphs (such as, paths, stars, complete graphs, DeBruijn graphs, hypercubes, Stirling graphs, etc.), RBG’s could also be used as interconnection networks.

1. INTRODUCTION

We define a family of labeled, undirected graphs based on the binary representation of positive integers as follows. A reverse binary graph of n nodes, denoted by RBG(n), has nodes labeled 1,2,. . . , n, and has an edge between node i and node j, for n > i > j 2 1, if and only if the usual binary representation of integer i has a 1 at its (j - l)th bit-position (counting the rightmost bit as the Oth bit). Figure 1 shows the first seven reverse binary graphs and their adjacency matrices. In the lower triangular portion of the adjacency matrix RA(n) of graph RBG(n), the jth row, when viewed as a string, is indeed the reverse of the binary string representing integer j, where 3<j<n.

The reverse binary graphs are incrementally generated in the sense that in graph RBG(n + 1))

the subgraph induced by nodes 1,2,. . . , n is RBG(n). These graphs are succinctly represented [3], that is, the adjacency matrix need not be stored explicitly to determine quickly (in time O(log n)) if two arbitrary nodes i and j are adjacent. We study several properties of these graphs. Table 1 summarizes several parameters of graph RBG(n). We obtain these parameters analytically in Section 3. Note that determining some of these parameters is NP-complete for general graphs [4]. Thus, this family of graphs can be used as test inputs to graph algorithms and heuristics. In [5], we have studied the directed version of RBG’s which are formed by the lower triangular portion of the adjacency matrix RA(n).

Several families of graphs, such as, paths, stars, complete graphs, DeBruijn graphs [6], hypercubes, Pascal graphs [7], R encontres graphs [8], Stirling graphs [9], etc., could be used as interconnection networks. Due to certain properties of reverse binary graphs, such as, incremen- tal generation, succinct representation, small diameter, small domination number, etc., RBG’s could also be used as interconnection networks.

We have used the INGRID software system developed by Brigham and Dutton [1,2] to study RBG’s. This system is designed to bound several graph invariants once knowledge of a few are available. Its theorem truing feature was specially helpful in developing some proofs. ‘This work is supported by Florida High Technology and Industry Council, by the U. S. Naval Training Systems Center, and by Texas Advanced Research Program Grant TARP-003594003.

Typeset by &+5-m

49

50 S.K. DAS et al.

PI

0 0 [I 0 cl 0 0 1 [ 1 0 01 110

0010 0010

[ 1 1101 0010

-0 01 0 1 d 00100~ 110111 001000 1 0 10 0 8 DllOO.

0

a 0

1 2

3

T

1 2

3

4

1 2

3 w 5 4

1 2

3

m 5 4 6

Figure 1. Some reverse binary graphs and their adjacency matrices.

The paper is organized as follows. Section 2 contains definitions and notations used in this paper. Section 3 studies the properties of RBG’s. Section 4 concludes the paper.

2. DEFINITIONS AND NOTATIONS

Standard graph theoretic terms used here can be found in textbooks such as [lO,ll]. An (undirected) edge between node i and node j will be denoted by < i, j >. All logarithms are

to the base 2. Integer log* n is the smallest k such that the kth iterated logarithm log@) n 5 1, where log(“) n = log(log tk-‘) n) and log(‘) - n - n. As usual, by 8 we mean “order exactly”, by R, “order no less than”, and by 0, “order no greater than.” A graph G is said to be perfect if G and each of its induced subgraphs have the property that the chromatic number equals the size of a maximum clique (12, p. 551. We adopt notations from [l] for various graph invariants of RBG’s. We list the invariants here with their symbols and provide definitions whenever appropriate.

(1) e = number of edges. (2) 6 = minimum node-degree. (3) A = maximum node-degree. (4) w = maximum clique size. (5) x = chromatic number. (6) xi = chromatic index (edge chromatic number).

Reverse binary graphs

Table 1. Some parameters of RBG(n)

51

(7)

(8)

(9)

(10) (11)

(12)

(13)

(14 (15) (16)

(17)

(18)

(19)

(20)

Parameter Value

number of edges EI(n logn)

minimum degree 1, for n 2 3

maximum degree @(n + log log n)

diameter O(log’ n)

girth 3, for n 2 5

circumference Iz(log* n), O(logn)

maximum clique size @(log’ n)

chromatic number @(log* n)

chromatic index equal to maximum degree, for n > 2g

domination number LlognJ - [log( LlognJ + 2)J + 1, for n 2 4

node independence number n - Llognj - 1, for n 2 16

matching number [logn] + 1, for n 2 16

node covering number LlognJ + 1, for n 2 16

edge covering number n - LlognJ - 1, for n 2 16

node clique cover number n - LlognJ - 1, for n 2 16

edge clique cover number 2 n - LlognJ - 1, for n > 16

node arboricity @(log’n) edge arboricity Wlogn), O(-Jnlogn) genus @(n log n) thickness WXn), O(Jnlogn)

or, = node covering number, i.e., the minimum number of nodes required to cover all the edges of RBG(n). C_Q = edge covering number, i.e., the minimum number of edges required to cover all the nodes of RBG(n). ,& = node independence number, i.e., the maximum number of nodes no two of which are adjacent. ,6i = matching number, i.e., the maximum number of edges no two of which are adjacent. u = domination number, i.e., the minimum number of nodes in a set such that any node not in the set is adjacent to some node in the set. &, = node clique cover number, i.e., the minimum number of cliques which include all the nodes of RBG(n). 81 = edge clique cover number, i.e., the minimum number of cliques which include all the edges of RBG(n) .

g = girth, i.e., the length of the shortest cycle in RBG(n). c = circumference, i.e., the length of the longest cycle in RBG(n). y = genus, i.e., the minimum “number of handles” required on a sphere to embed RBG(n) without any edge-crossings. a = node arboricity, i.e., the minimum size of a partition of the nodes of RBG(n) such that each subset of nodes in the partition induces an acyclic subgraph of RBG(n). al = edge arboricity, i.e., the minimum size of a partition of the edges of RBG(n) such that each subset in the partition is an acyclic subgraph of RBG(n). t = thickness, i.e., the minimum number of planar subgraphs whose union is the entire graph RBG(n). X = spectral index, i.e., the largest eigenvalue of the adjacency matrix RA(n) of graph RBG(n).

3. SOME PROPERTIES OF REVERSE BINARY GRAPHS

By definition, matrix RA(n) is a submatrix of RA(n + 1). Hence, graph RBG(n) is a subgraph of RBG(n + 1) induced by the node set {1,2,. . . ,n}. Thus, these graphs can be generated recursively. That is, RBG(n + 1) can be obtained by simply adding node (n + 1) to RBG(n)

52 S.K. DAS et al.

and joining it with appropriate edges. The new node added to RBG(n) to form RBG(n + 1) is connected to the rest of the graph via the new edge < n + 1, [log(n + l)] + 1 >. Therefore, RBG(n) is connected, for n > 3. Also, since a node i is adjacent to node ([log iJ + l), for i 2 3, the sequence of nodes

i, [logi] + 1, [log( Llogij + l)] + 1,. . . ,3,2

is a path in RBG(n). Thus, the distance between any node and node 3 is O(log* n). Hence, the diameter of RBG(n) is O(log* n). Since node 211°snl is connected to the rest of the graph by the lone edge < 211°snl, [lognj + 1 >, for n > 4, the minimum degree of nodes, 6(n), in RBG(n) is one, for n 2 4. By definition, for the first three RBG’s, 6(l) = 6(2) = 0, and S(3) = 1. As a consequence, the node and the edge connectivity of RBG(n) are both one, for n 2 3.

Nodes 1, 3 and 5 form a cycle in RBG(5). Therefore, the length of the shortest cycle in RBG(n), i.e., the girth of RBG(n), is g(n) = 3, for n > 5. The presence of this odd length cycle implies that RBG(n) is a non-bipartite graph. The following theorem shows that almost all reverse binary graphs are nonplanar.

THEOREM 1. Graph RBG(n) is nonplanar for n > 15.

PROOF. Figure 2 depicts a planar embedding of RBG(14). In order to construct RBG( 15), we add node 15 and edges < 15,l >, < 15,2 >, < 15,3 >, and < 15,4 > to RBG(14). Figure 3 shows a subgraph of RBG(15). Cl early, it can be homeomorphically obtained from the complete bipartite graph Ks,s. Therefore, graph RBG(15) is nonplanar. Since RBG(15) is a subgraph of every graph RBG(n) for n > 15, the theorem follows. I

Figure 2. A planar embedding of graph RBG( 14).

Figure 3. A subgraph of RBG(15).

We now calculate the number of edges, e(n), in the graph RBG(n). Integer e(n) is given by the number of l’s in the lower (or, the upper) triangular portion of the adjacency matrix RA(n) of graph RBG(n). Let us obtain a new matrix RA’(n) from RA(n) by setting rii = r22 = 1, and rij = 0 for 1 5 i < j 5 n. Let the number of l’s in RA’(n) be e’(n). Clearly, the total number of edges in RBG(n) is

e(n) = e’(n) - 2, for n > 3. (I)

LEMMA 1. e’(2k - 1) = k2”-‘, fork > 1.

PROOF. The first k columns of RA’(2k - 1) contain 2”-l number of l’s each and the rest of the columns contain all 0’s. I

Reverse binary graphs 53

COROLLARY 1. e’(2”) = k2k-1 + 1, for k 2 0.

PROOF. Since (2”)th row in RA’(2k) contains a single 1, e’(zk) = e’(2k - 1) + 1, for k 2 1.

Furthermore, e’(2O) = 1, by definition of RA’(2’). I

COROLLARY 2. The lower and the upper bound on the number of edges in RBG(n), for n > 3, is given by the relation,

~lognJ21’ognJ-1 - 1 5 e(n) 2 (Llognj + 1)211°gnJ - 2.

PROOF. For n > 3, 2k 5 n 5 2 k+l - 1, where k = Llog n]. Therefore, e(2k) < e(n) 5 e(2”+i - 1). Clearly, for n 2 3, k 1 2. Therefore, using Lemma 1, Corollary 1, and Expression (l), we get, k2k-1 - 1 5 e(n) 5 (k + 1)2k - 2, for n > 3. I

Corollary 2 implies that the number of edges in graph RBG(n) is Q(n logn). To get an exact expression, we still need the following two lemmas.

LEMMA 2. e’(2k + i) = e’(2k) + e’(i) $ i, for 1 5 i < 2”, and k 1 0.

PROOF. Matrix RA’(2k + i) for 1 L i < 2k is shown in Figure 4. Its partitioning into several submatrices leads to the lemma. I

1 2 --- k k+l k+2 - - .

1

2

i

pk-l

2k 2k+l

2k+2

2k+i

1 0 --- 0

0 1 --- 0

0

0

_ _ ___ _ _ _____ bl b2 --- E 0 - ----- -

- --.-- -

- --.-. -

1 l---l 0

_ _-_

0 0 _ - _

1 0 --- 01

0 1 --- 0 _ _ -.- -

_ _ 0-e -

_ _ ___ -

bl b2 - - - bk

1 1

1 I 1

_ _-_ 0 _ _ _

_ _-_

_ _-_

_ ___

0 _ _ _

0 _ - _

0 _ - _

0 - - _

_ .-_

_ ___

Figiure 4. Matrix RA’(2k + i).

2k,i

0

0

0

0

0

0

0

0 _

LEMMA 3. Let n = 2”l + 2k1-1 + ... + 2k2 + 2k1, where kl > kl_l > .. . > ICI. Then,

e’(n)=‘$ (e'(2ka)+g2kj) =$ bk* (l-i+;)] $1.

PROOF. Consider n = 2kl + (2kc-l + 2’[--2 + . . . + Zkl). Applying Lemma 2, we have

l-1

e'(n) = e'(zkl) + e'($1 + 2k1-2 +... + $1) +x24

j=l

z-2 l-l

= e'(Zki) + e'(2kl-1) + e'(2kI-2 + . . . + 2kl) + C2kj + C2kj j=l j=l (2)

1 a-1

= x( e'(2k*) + C 2"j . i=l j=l 1

54 S.K. DAS et al.

Now, applying Corollary 1 to Expression (2) and manipulating algebraically, we have

e’(n) = e (ki2kt-1 + 1) + 2 ((I - i)2”9 = & [2*. (1 -i + s>] + 1. 1 i=l i=l i=l

THEOREM 2. Let n = 2k1+2kL-1+...+2k2+2k1, wherekl > kl_1 > ... > kl. Then, thenumber of edges in graph RBG(n) is

PROOF. The proof follows from Lemma 3 and Expression (1).

For example, since 5 = 22 + 2’, we have 1 = 2, ki = 0, and k:! = 2. Therefore,

I

e(5) = 2”’ (l-,.+).,q-2++-2

=,.(2-I.9 +22(2-2+;)+2-2

= 5.

Next, we determine the maximum degree, A(n), of RBG(n).

THEOREM 3. The maximum degree of RBG(2” - 1) is

A(2” - 1) = 2”-l + [log(k + l)j, for k > 3.

PROOF. The degree of a node i in graph RBG(2” - l), for 1 5 i < 2k - 1, is given by the number of l’s in the ith row and the ith column of the lower triangular portion of the adjacency matrix RA(2” - 1). In this portion of matrix RA(2” - l), columns 3 through k have 2”-l number of l’s each, and columns (k + 1) through n have all 0’s. Clearly, the highest degree node is among nodes 1 through k. Now, the row i < k with the maximum number of l’s is the largest integer i of the form 2l - 1 5 k. This row is (21’0s(k+1)l - 1) containing Llog(k + l)] number of 1’s. Integer (2llos(k+i)J _ 1) 1s at least 3 for k > 3. Thus, node (21’“s(k+1)J - 1) is one among the highest degree nodes in RBG(2’ - 1) and its degree is (2”-i + jlog(k + l)]). I

COROLLARY 3. A(2k) = A(2” - l), for k > 3.

PROOF. The addition of node 2k in graph RBG(2k - 1) to form RBG(2”) has the sole effect of increasing the degree of node (k + 1) by one. Since the column (k + 1) in the lower triangular portion of matrix RA(2k) contains a single 1, the degree of node (k + 1) is bounded from above by (1 + [log(k + l)]), for k 2 3. Thus, node (2L10s(k+1)J - 1) continues to be among the nodes

with the maximum degree. I

The following corollary gives the lower and the upper bound on the maximum degree of graph RBG(n).

COROLLARY 4.

2110gnl-1 + Llog([lognj + 1)J I: A(n) < 211°gnl + Llog(llogn] +2)J, forn > 8.

PROOF. For n > 8, 2k I n I 2k+1 - 1, where k = llog n] . Since k > 3 for n > 8, Corollary 4 follows by applying Theorem 3 and Corollary 3. I

Thus, the maximum degree of RBG(n) is O(n + log log n). We now turn our attention to some graph invariants related to coloring. We first determine the

maximum clique size, w(n), in RBG(n). Later, we show that the chromatic number of RBG(n) is equal to its maximum clique size. Notice that RBG(5) is the smallest RBG to contain a clique of


size 3 (see Figure 1). Similarly, RBG(3) and RBG(l) are the smallest RBG’s to contain cliques of size 2 and 1, respectively. Further reasoning along this line results in the following lemma.

LEMMA 4. Let n1 = 1, n2 = 3, n3 = 5, and ni = 2(na-1-‘) + ni-1, for i 2 4. Then, ni is the smallest integer such that RBG(ni) contains a clique of size i, i 2 1.

PROOF. (by induction) Basis: The lemma holds for nl, n2, and ns by inspecting the graphs RBG(j), 1 < j 5 5. Induction: Assuming that the lemma holds for n1 through nk, k > 3, we show that it holds

for nk+l also. Since nk is the smallest integer such that RBG(nk) contains a clique of size k, node nk must belong to this clique. Also, since nk+l = 2nk-1 + nk, node nk+l in RBG(nk+l) is adjacent to node nk and all the nodes that nk is adjacent to. Thus, the clique of size k in RBG(nk) together with node nk+l forms a clique of size (k + 1) in RBG(nk+l).

We complete the induction step by showing that nk+l is the smallest such integer. The proof is by contradiction. Let 2 be the smallest integer smaller than nk+l such that RBG(x) has a clique of size (k + 1). Since x < nkfl and the binary representation of nk+l has l’s only at the bit positions (ni - l), for 1 < i 5 k, there exists a j such that the binary representation of z has a 0 at the bit position (nj - l), for 1 < j 5 k, and x matches with nk+l at every higher significant bit position 1, for (nk - 1) > 1 > nj. Therefore, node x in RBG(x) is adjacent to only (k - j) nodes with labels greater than or equal to nj. Since x is adjacent to all the k nodes in the clique of size (k + 1) in RBG(x) ( as x is the smallest such integer), there are at least j nodes of this clique labeled less than nj. These j nodes, therefore, constitute a clique of size j among themselves. However, this implies that nj was not the smallest integer such that RBG(nj) contains a clique of size j, where 1 5 j < k. This leads to a contradiction. I

COROLLARY 5. The chromatic number of graph RBG(ni), x(ni) 2 i, for i > 1.

THEOREM 4. x(ni) = i, for i 2 1.

PROOF. x(ni) 2 i, for i > 1, from Corollary 5. We further prove that x(ni) 5 i, using induction on i.

Basis: By inspection, x(ni) < i, where i = 1,2, and 3. Induction: let us assume that x(ni) < i, for i 5 k. Hence, RBG(nk) can be colored using colors

1 through k. Since nodes (nk + 1) through nk+l are adjacent only to nodes 1 through nk in RBG(nk+l), each of nodes (nk + 1) through nk+l can be assigned the color (k + 1). I

Due to Lemma 4 and Theorem 4, the smallest reverse binary graph requiring i colors for a proper coloring of its nodes is RBG(ni), f or i > 1. Furthermore, for an arbitrary n, RBG(n) has w(n) = x(n) = @(log* n). Table 2 gives the chromatic numbers of a few RBG’s. Since the maximum clique size and the chromatic number of RBG( ) n are equal, one is tempted to show that these graphs are perfect graphs. However, as the following theorem shows, RBG’s are not perfect graphs, in general.

Table 2. The chromatic numbers of some RBG’s.

ni x(w) 711 = 2’ = 1 1

7x2 = 21 + 20 = 3 2

ns = 22 + 20 = 5 3

n* = 24 + 22 + 20 = 24 + 5 = 21 4

ng = 220 + 2* + 22 + 2O = 220 + 21 5

ns = 2n5-1 + ng = 2220+20 + 220 + 21 6

THEOREM 5. RBG(n) is not a perfect graph for n 1 129.

PROOF. The subgraph induced by nodes 1, 3, 4, 8 and 129 in RBG(129) is a cycle without chords. Since the maximum clique size and the chromatic number of a cycle of five nodes are two and three, respectively, RBG(129) is not a perfect graph. Due to subgraph property, RBG(n) is also not a perfect graph for n > 129. I

56 S.K. DAS et al.

Next, we show that the chromatic index is equal to the maximum degree for almost all RBG’s. For this purpose, we first establish an upper bound on the spectral index, X(n), of RBG(n), with respect to its maximum degree, A(n).

LEMMA 5. The spectral index of RBG(n), A(n) 5 %$, for n 2 2g.

PROOF. Using Theorem 10 of [13, p. 621, we have x(n) < J2e(n)(l - l/x(n)). This relation gives

X(n) F m. (3)

Using Corollary 2, we have e(n) I 2”(K + 1) - 2, where k = [IognJ. This gives e(n) 5 2”(k+ 1). Therefore, Expression (3) yields

x(n) < J/G. (4)

Next, by Corollary 4, the maximum degree of RGB(n), A(n) > 2”-l + Llog(k + l)]. Thus,

a(n) > 2k-2 2- ’ (5)

after dropping the second term. Now, since dm 5 2k-2, for k 2 9, we get, from Expressions (4) and (5), x(n) I (A(n)/2). I

THEOREM 6. The chromatic index xl(n), of RBG( ) n is equal to its maximum degree, A(n), for n 2 2g.

PROOF. As shown in Corollary 1 of [14, p. 381, if x(n) I (A(n)/2), then xl(n) = A(n). Therefore, Theorem 6 follows from Lemma 5.

Now, we explore independence and covering properties of RBG’s. The next two results determine the domination number of RBG(n).

LEMMA 6. The domination number of RBG(n),

o(n) 2 LlognJ - Llog( [logn] + 2)] + 1, for 2 1.

PROOF. The number of significant bits required to represent integers 1 through n is s = LlognJ + 1. In RBG(n), nodes (s+l) through n form an independent set of nodes. Among these, consider node (s + l), and the nodes of the form 2” such that s + 2 5 2” 5 n. That is rlog(s + 2)l < k 5 llognl, or [log(s + 1)J + 1 I k I [lognj. Since node (s + 1) is adjacent only to node

(llog(s + I)1 + 1) an d zero or more of nodes 1 through llog(s + l)], at least one of the nodes in theset {(s+1),1,2,3 ,..., [log(s+l)]+l} must belong to a dominating set of graph RBG(n) in order to dominate node (s + 1). Next, node 2L’0g(s+1)l+1 is adjacent only to node ([log(s+l)] +2). Therefore, at least one of these two nodes must be in a dominating set of RBG(n). The same argument holds for each of the nodes 2 k, for llog(s + l)] + 1 5 k < llognj. Therefore, a dominating set of RBG( n must contain at least [l + (LlognJ - (llog(s + l)] + 1) + l)] nodes. ) This simplifies to yield

a(n) > L(logn] - llog( [logn] + 2)] + 1, for n 2 1. I

THEOREM 7. The domination number of graph RBG(n) is given by,

g(n) = LlognJ - [log( LlognJ + 2)J + 1, for n 1 4.

PROOF. We claim that the set

DOM = { llog(s + l)] + 1, [log(s + l)] + 2,. . . , s},

where s = [logn] + 1, is a dominating set of RBG(n), for n > 4.


Since (2L’0s(S+1)J+1 - 1) 2 (s + 1) 2 2L’0s(S+1)J for n 2 4, node ([log(s + l)] + 1) dominates, besides itself, nodes (s + 1) through (2 L1og(s+l)l+l - 1). Furthermore, node ([log(s + l)] + 2) dominates nodes 2 L1Og(S+l)J+l through (2 L’Og(s+l)J +2 - 1)) node ( [log( s + 1)J + 3) dominates nodes 2llOJX(S+1)1+2 through (2llOEds+l)l+3 _ I), and so on, and, finally, node s = LlognJ + 1 dominates nodes 2L1”gnJ through n. We have, thus, accounted for nodes ([log(s + l)] + 1) through n. The remaining nodes 1 through [log(s + l)] are dominated by node (2L“‘g(s+1)J - 1). The latter node belongs to set DOM because ([log(s + 1)j + 1) < (2L’“g(a+1)J - 1) < s, for s 2 3. Thus, DOM is a dominating set of RBG(n), and its size is

1 DOMl = s - (llog(s + l)] + 1) + 1

=~lognJ+1-~log(~logn~+2)J-1+l

= LlognJ - llog( LlognJ + 2)J + 1.

This, coupled with Lemma 6, completes the proof. I As an example, a smallest dominating set (of size 4) in graph RBG(50) is {3,4,5,6}. The next

theorem gives the matching number of RBG(n).

THEOREM 8. The matching number of RBG(n), 01(n) = LlognJ + 1, for n 2 16.

PROOF. Nodes (s + 1) through n, where s = LlognJ + 1, form an independent set of nodes in RBG(n). Since s 5 n - s for n > 16, a matching in RBG(n) can contain a maximum of s edges, one incident to each of nodes 1 through s. Thus, ,01(n) 5 s.

We now describe a maximum matching of RBG(n) containing s edges. The edges are < 1, 29-2 + 20 >, < 2, 25-2 + 21 >, < 3, 2S-2 + 22 >, . . . ) <s-2, 25-2+29--3>,<s-1, 25-2>, and < s, 2S-1 >. All these edges are distinct and non-adjacent because

1 < 2 < . .. < s < 2*-2 < (2S-2 + 2’) < ... < (2S-2 + 2s-3) < 2’-’ 2 71,

for n > 16. Thus, PI(n) = s = llogn] + 1, for n 2 16. I

According to Theorem 8, a maximum matching in graph RBG(50) consists of the edges < 1,17 >,< 2,18 >,< 3,20 >,< 4,24 >,< 5,16 >, and < 6,32 >. Theedge covering number, al(n), of the graph RBG(n) is easily calculated using Theorem 8 as follows.

COROLLARY 6. The edge covering number of RBG(n), al(n) = (n - llognl - l), for n 2 16.

PROOF. This follows from the fact that &l(n) +/31(n) = n, for a connected graph [ll, p. 951. 1

Next, we obtain an expression for the node independence number of RBG’s.

THEOREM 9. The node independence number of RBG(n), PO(n) = (n - LlognJ - l), for n > 16.

PROOF. If a graph G of n nodes has a matching of size PI, then any independent set of nodes in G can include only one end-node of each of the p1 edges of the matching, in addition to a maximum of n - 2p1 nodes which are not incident on any of the edges in the matching. Thus, its node independence number PO 5 n - PI. Therefore, from Theorem 8, we conclude that

PO(n) 5 n - llognj - 1, f or n 2 16. We already know that nodes (s + 1) through n form an independent set of nodes in RBG(n), where s = LlognJ + 1. Therefore, PO(n) = n - [log nJ - 1,

for n 2 16. I

Using Theorem 9 and the fact that so(n) + PO(n) = n f or any nontrivial connected graph [ll, p. 951, the node covering number of RBG(n) follows.

COROLLARY 7. The node covering number of graph RBG(n) is given by,

ctg(n) = (llognl + l), for n 2 16.

The following theorem determines the node clique covering number of RBG(n).

THEOREM 10. The node clique covering number of RJ3G(n), e,-,(n) = (n- LlognJ -l), for n > 16.

PROOF. The node clique covering number, &,(n), of a graph is bounded from below by its node independence number. Furthermore, 00(n) can not exceed the edge covering number of a graph [l, Rule 2121. Th ere ore, f the result follows from Corollary 6 and Theorem 9. I

58 S.K. DAS et al.

COROLLARY 8. The edge clique cover number of RBG(n), 0,(n) > n - [logn] - 1, for n 2 16.

PROOF. It follows directly because @l(n) > e,(n). I

We now use the properties studied so far to establish lower and upper bounds on some other parameters of RBG’s. The following theorem relates to the circumference of RBG(n).

THEOREM 11.

(1) The circumference, c(n), of RBG(n) is greater than or equal to the size of its maximum clique, w(n).

(2) c(n) I 2LlognJ + 4, for n 2 16.

PROOF.

(1) Since x(n) = w(n) for RBG(n), the proof follows from Proposition 2 of [15, p. 61 according

to which c(n) 2 “‘(‘$$~)l-ll.

(2) From Rule 79 of [l], we have Lc(n)/2] I ,&(n). Th is relation yields c(n)/2 < p,(n) + 1, which further simplifies to c(n) i 2pl(n) + 2. Substituting ( LlognJ + 1) for PI(~) from Theorem 8, we get c(n) I 2llognJ + 4, for n 2 16. I

Theorem 11 establishes a lower bound of 0(log* n) and an upper bound of O(logn) on the circumference of RBG(n). The following theorem provides bounds on the genus, r(n), of RBG(n).

THEOREM 12. The genus of RBG(n) is exactly of the order n log n. That is, r(n) = @(n log n).

PROOF. We prove this by showing both a lower and an upper bound on r(n). Since the girth,

s(n), of RBG( ) n exists and its node connectivity is greater than zero for n 2 5, we have, according

to [16, P. 961, r(n) L 9(1-&)-q fl. Since g(n) = 3, we get r(n) 2 e(n)/6-n/2+1. This

relation gives a lower bound on genus of RBG(n) as r(n) = R(n log n), because e(n) = @(n log n) by Corollary 2. Next, we have the following relation for a connected graph [17, p. 2291: r(n) 5

(e(n) - n + 1) (+ - 410g~e(~~_-n+l~) . Therefore, r(n) =)(nlogn). I

The following two theorems pertain to determining node and edge arboricity of RBG’s.

THEOREM 13. The node arboricity, u(n), of RBG(n) is exactly of the order log* n. That is, a(n) = @(log* n).

PROOF. Using Rule 51 of [l], u(n) > x(n)/2. Next, from Rule 53 of [l],

Since girth g(n) = 3, for n > 5, and chromatic number x(n) = @(log* n), Theorem 13 follows. 1

THEOREM 14. The edge arboricity, al(n), of RBG( n is at least of the order logn and at most ) of the order dz. That is, al(n) = Sl(logn) and al = O(dm).

PROOF. For the lower bound on al(n), we use Rule 379 of [2], according to which, al(n) 2 s, for a connected graph. For the upper bound, we use Theorem 8 of [18], which gives al(n) 5

r-1 3y( ) + 2, for a nonplanar graph. Theorem 14 follows directly because RBG(n) is nonplanar

for n 2 15, the number of edges e(n) = O(nlogn) and its genus r(n) = O(nlogn). I

COROLLARY 9. The thickness, t(n), of RBG( n ) is at least of the order log n and at most of the order d-.

PROOF. This follows from Rules 372 and 380 of (21 which provide tight bounds on thickness as

al(n)/3 I t(n) < al(n). I


4. CONCLUSION

The reverse binary graphs, defined simply using the binary representation of positive integers, have turned out to possess an interesting mathematical structure. We have studied several properties of these graphs and determined a number of its parameters, including some which are NP-complete for graphs in general. Therefore, this family of graphs can be used as test inputs for graph algorithms and for comparing various approximation algorithms for these NP- complete problems. The RBG’s could also be used as interconnection networks. Table 3 compares some parameters of graph RBG(n) with that of hypercube of order n, for n a power of 2 (the domination number of hypercube graph as shown in Table 3 is known only when the number of nodes n = 22’-1, for 1 2 0 [19]). W e remark that although hypercubes are regular and RBG’s are not, a hypercube graph can be incremented only by doubling its size, whereas an RBG(n) can be incremented to RBG(n + 1) by adding a single node.

In this paper, we have established only lower and upper bounds on some parameters of RBG’s, such as edge arboricity and circumference. Further work can be aimed at the derivation of exact expressions for these parameters and study of other parameters not considered here. It would also be worthwhile to investigate other application areas of RBG’s.

1.

2.

3. 4.

5.

6.

7.

a.

9.

10.

11. 12.

13.

14.

Table 3. Comparison of RBG and hypercube.

Parameter RBG Hypercube

number of edges $logn- 1 qlogn

diameter O( log’ n) log n

node connectivity 1 log 71

edge connectivity 1 log n

chromatic number O(log’n) 2

domination number Q(logn) n lognfl

node independence number n-logn-1 $

matching number logn + 1 14 2

genus @(n logn) O(n log n)

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N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall, Englewood Cliffs, NJ, (1974). F. Harary, Graph Theory, Addison-Wesley, Reading, MA, (1969). L. Lov&z, Perfect graphs, In Selected Topics in Graph Theory, (Edited by L.W. Beineke and R.J. Wilson), Vol. II, pp. 55-87, Academic Press, London, (1983). C.S. Edwards and C.H. Elphick, Lower bounds for the clique and the chromatic numbers of a graph, Discrete Appl. Math. 5, 51-64 (1983). S. Fiorini and R.J. Wilson, On the chromatic index of a graph: II, In Combinatorics, London Math Sot. Lecture Note Ser., No. 13 pp. 37-51, Cambridge Univ. Press, London, (1974).

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