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Independence in direct-product

graphs

Pranava K Jha

Dept of Computer Engineering

Delhi Institute of Technology: Delhi

Kashmere Gate, Delhi 110 006, INDIA

e-mail: pkj@dit.ernet.in

Sandi Klavˇ zar∗

Department of Mathematics, PEF

University of Maribor

Koroˇ ska cesta 160, 62000 Maribor, SLOVENIA

e-mail: sandi.klavzar@uni-lj.si

Abstract

Let α(G) denote the independence number of a graph G and let

G × H be the direct product of graphs G and H. Set α(G × H) =

max{α(G) · |H|, α(H) · |G|}. If G is a path or a cycle and H is a

path or a cycle then α(G×H) = α(G×H). Moreover, this equality

holds also in the case when G is a bipartite graph with a perfect

matching and H is a traceable graph. However, for any graph G

with at least one edge and for any i ∈ IN there is a graph H such

that α(G × H) > α(G × H) + i.

1Introduction

Problems of determining the independence number and the matching num-

ber of a graph are also important because of applications of these invariants

in many areas, notably, (i) selection by PRAM, (ii) VLSI layout, (iii) wire

coloring, and (iv) processor scheduling. While independence number prob-

lem is NP-hard [5], matching is solvable in polynomial time [17]. Recently

it was shown that independence number is not even approximable within a

∗This work was supported in part by the Ministry of Science and Technology of

Slovenia under the grant J1-7036.

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factor of ncfor any c > 0 unless P=NP [2, 3]. For a product graph, solving

these problems via factor graphs is economical, since problem size is much

smaller in the factors than in the product. This natural view forms the

basis of several studies, for example, [7, 13, 19].

The present paper addresses the twin problems with respect to the

direct product. The direct product G×H of graphs G and H is a graph with

V (G×H) = V (G)×V (H) and E(G×H) = {{(u,x),(v,y)}|{u,v} ∈ E(G)

and {x,y} ∈ E(H)}. This product (which is also known as Kronecker

product, tensor product, categorical product and graph conjunction) is the

most natural graph product. It is commutative and associative in a natural

way. However, dealing with this product is also most difficult in many

respects among standard products. For instance, a Cartesian product or a

strong product of two graphs is connected if and only if both factors are

connected, and this fact is easily provable. On the other hand, it is not

completely straightforward to see that G × H is connected if and only if

both G and H are connected and at least one of them is non-bipartite,

cf. [20]. Furthermore, if both G and H are connected and bipartite, then

G × H consists of two connected components.

The direct product has several applications, for instance it may be used

as a model for concurrency in multiprocessor systems [15]. Some other

applications are listed in [12].

By a graph is meant a finite, simple, undirected graph. Unless indicated

otherwise, graphs are also connected and have at least two vertices. Let

|G| stand for |V (G)|. For X ⊆ V (G), ?X? denotes the subgraph induced

by X. By χ(G), α(G) and τ(G) we will denote the chromatic number,

the independence number and the matching number of G, respectively. A

graph has a perfect matching if τ(G) = |G|/2. If G is a bipartite graph

with V (G) = V0+ V1and |V0| ≤ |V1| then a complete matching from V0to

V1is a matching which includes every vertex of V0.

The main open problem concerning the direct product is the Hedet-

niemi’s conjecture. Let χ(G × H) = min{χ(G), χ(H)}. It is easily seen

that χ(G×H) ≤ χ(G×H) holds for any graphs G and H. In 1966, Hedet-

niemi [9] conjectured that for all graphs G and H, χ(G×H) = χ(G×H).

For surveys on the conjecture we refer to [4, 14]. The chromatic number of a

graph G and its independence number are closely related via the inequality

χ(G) ≥ |G|/α(G). It is easy to see (and well-known [13, 18]) that

α(G × H) ≥ max{α(G) · |H|, α(H) · |G|} =: α(G × H).

The main topic of this paper is the study of relation between α(G × H)

and α(G × H). For instance, is it the case (analogous to the Hedetniemi’s

conjecture) that α(G × H) = α(G × H) for any two graphs G and H ? In

particular, does α(G × Cn) = α(G × Cn) hold for any graph G?

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[15] R. H. Lamprey and B. H. Barnes, Product graphs and their appli-

cations, Modelling and Simulation, 5 (1974) 1119–1123 (Proc Fifth

Annual Pittsburgh Conference, Instrument Society of America, Pitts-

burgh, PA, 1974).

[16] F. Lazebnik, V. A. Ustimenko and A. J. Woldar, A new series of

dense graphs of high girth, Rutcor Research Report 33–93, Rutgers

University, NJ, Dec 1993.

[17] J. A. McHugh, Algorithmic Graph Theory, Prentice–Hall, 1990.

[18] R. J. Nowakowski and D. Rall, Associative graph products and their

independence, domination and coloring numbers, manuscript, June

1993.

[19] E. Sonnemann and O. Kraft, Independence numbers of product

graphs, J. Combin. Theory Ser. B 17 (1974) 133–142.

[20] P. M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math.

Soc. 13 (1962) 47–52.

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