Linear Coloring and Linear Graphs

arXiv:0807.4234v1 [cs.DM] 26 Jul 2008 Linear Coloring and Linear Graphs ∗ Kyriaki Ioannidou and Stavros D. Nikolopoulos Department of Computer Science, University of Ioannina P.O.Box 1186, GR-45110 Ioannina, Greece {kioannid, stavros}@cs.uoi.gr Abstract: Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology [9], and the framework through which it was studied, we introduce the linear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be linearly colored in polynomial time by proposing a simple linear coloring algorithm. Based on these results, we define a new class of perfect graphs, which we call co-linear graphs, and study their complement graphs, namely linear graphs. The linear coloring of a graph G is a vertex coloring such that two vertices can be assigned the same color, if their corresponding clique sets are associated by the set inclusion relation (a clique set of a vertex u is the set of all maximal cliques containing u); the linear chromatic number λ(G) of G is the least integer k for which G admits a linear coloring with k colors. We show that linear graphs are those graphs G for which the linear chromatic number achieves its theoretical lower bound in every induced subgraph of G. We prove inclusion relations between these two classes of graphs and other subclasses of chordal and co-chordal graphs, and also study the structure of the forbidden induced subgraphs of the class of linear graphs. Keywords: Linear coloring, chromatic number, linear graphs, co-linear graphs, chordal graphs, co-chordal graphs, strongly chordal graphs, algorithms, complexity. 1 Introduction Framework-Motivation. A linear coloring of a graph G is a coloring of its vertices such that if two vertices are assigned the same color, then their corresponding clique sets are associated by the set inclusion relation; a clique set of a vertex u is the set of all maximal cliques in G containing u. The linear chromatic number λ(G) of G is the least integer k for which G admits a linear coloring with k colors. Motivated by the definition of linear coloring on simplicial complexes associated to graphs, first introduced by Civan and Yal¸cin [9] in the context of algebraic topology, we define the linear coloring on graphs. The idea for translating their definition in graph theoretic terms came from studying linear colorings on simplicial complexes which can be represented by a graph. In particular, we studied the linear coloring on the independence complex I(G) of a graph G, which can always be represented by a graph and, more specifically, is identical to the complement graph G of G in graph theoretic terms; indeed, the facets of I(G) are exactly the maximal cliques of G. However, the two definitions cannot always be considered as identical since not in all cases a simplicial complex can be represented by a ∗This research is co-financed by E.U.-European Social Fund (75%) and the Greek Ministry of Development-GSRT (25%). 1 graph; such an example is the neighborhood complex N(G) of a graph G. Recently, Civan and Yal¸cin [9] studied the linear coloring of the neighborhood complex N(G) of a graph G and proved that, for any graph G, the linear chromatic number of N(G) gives an upper bound for the chromatic number of the graph G. This approach lies in a general framework met in algebraic topology. In the context of algebraic topology, one can find much work done on providing boundaries for the chromatic number of an arbitrary graph G, by examining the topology of the graph through different simplicial complexes associated to the graph. This domain was motivated by Kneser’s conjecture, which was posed in 1955, claiming that “if we split the n-subsets of a (2n + k)-element set into k + 1 classes, one of the classes will contain two disjoint n-subsets” [16]. Kneser’s conjecture was first proved by Lov´asz in 1978, with a proof based on graph theory, by rephrasing the conjecture into “the chromatic number of Kneser’s graph KGn,k is k + 2” [17]. Many more topological and combinatorial proofs followed the interest of which extends beyond the original conjecture [21]. Although Kneser’s conjecture is concerned with the chromatic numbers of certain graphs (Kneser graphs), the proof methods that are known provide lower bounds for the chromatic number of any graph [18]. Thus, this initiated the application of topological tools in studying graph theory problems and more particularly in graph coloring problems [10]. The interest to provide boundaries for the chromatic number χ(G) of an arbitrary graph G through the study of different simplicial complexes associated to G, which is found in algebraic topology bibliography, drove the motivation for defining the linear coloring on the graph G and studying the relation between the chromatic number χ(G) and the linear chromatic number λ(G). We show that for any graph G, λ(G) is an upper b

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