Colinear Coloring on Graphs

Abstract

Abstract. Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology, and the framework through which it was studied, we introduce the colinear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be colinearly colored in polynomial time by proposing a simple algorithm. The colinear 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 colinear chromatic number λ(G) of G is the least integer k for which G admits a colinear coloring with k colors. Based on the colinear coloring, we define the χ-colinear and α-colinear properties and characterize known graph classes in terms of these properties.

Harmonious Coloring on Subclasses of Colinear Graphs

“…In [17] we presented a polynomial time algorithm for colinear coloring which can be applied to any graph G and, also, we proved the following results.…”

“…For basic definitions in graph theory refer to [4,15], and for more details on colinear coloring, colinear and linear graphs refer to [17,16].…”

“…In this paper we study the complexity status of the harmonious coloring problem on two subclasses of colinear graphs [17,16]. We first show that the harmonious coloring problem is NP-complete on split undirected path graphs and, then, we show that the class of split undirected path graphs forms a subclass of colinear graphs; thus, we obtain the NP-completeness of the harmonious coloring problem on colinear graphs as well.…”

“…Motivated by the definition of linear coloring on simplicial complexes associated to graphs, first introduced in the context of algebraic topology [8], we recently introduced the colinear coloring on graphs [17].…”

“…Moreover, we provide a polynomial solution for the harmonious coloring problem on split strongly chordal graphs, the interest of which lies on the fact that the problem is NP-complete on both split graphs and strongly chordal graphs [1,3]. However, the complexity status of the problem for the class of connected linear graphs still remains an open question; note that the harmonious coloring problem is NP-complete on disconnected linear graphs, since it is NP-complete on disconnected quasi-threshold graphs [2] and quasi-threshold graphs form a subclass of linear graphs [17,16].…”