Colinear Coloring on Graphs

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 αcolinear properties and characterize known graph classes in terms of these properties.