Relation between Cartesian product and adjacent vertex distinguishing coloring

Abstract

A proper k-edge coloring of a graph G is an assignment of k colors 1, 2, …, k to edges of G such that any two adjacent edges receive the different colors. For a proper edge coloring f of G and any vertex x of G, we use Sf(x) or S(x) to denote the set of the colors assigned to the edges incident with x. If for any two adjacent vertices u and v of G, we have S(u)≠S(v), then f is called the adjacent vertex distinguishing proper edge coloring of G (or AVDPEC of G in brief). The minimum number of colors required in an AVDPEC of G is called the adjacent vertex distinguishing proper edge chromatic number of G, denoted by χ'a(G). A proper k-total coloring of a graph G is an assignment of k colors 1, 2, …, k to vertices and edges of G such that any two adjacent or incident elements receive the different colors. For a proper total coloring g of G and any vertex x of G, we use Cg(x) or C(x) to denote the set of the colors assigned to the vertex x and edges incident with x. If for any two adjacent vertices u and v of G, we have C(u)≠C(v), then g is called the adjacent vertex distinguishing total coloring of G (or AVDTC of G in brief). The minimum number of colors required in an AVDTC of G is called the adjacent vertex distinguishing total chromatic number of G, denoted by χ″a(G). In this paper, we discuss the relation between Cartesian product and two types of adjacent vertex distinguishing coloring.