L(2,1)-colorings and irreducible no-hole colorings of the direct product of graphs

An L(2,1)-coloring (or labeling) of a graph G is a mapping f:V(G)→Z+⋃{0} such that |f(u)−f(v)|≥2 for all edges uv of G, and |f(u)−f(v)|≥1 if u and v are at distance two in G. The span of anL(2,1)-coloring is the maximum color (or label) assigned by it. The span of a graphG, denoted by λ(G), is equal to min{span f:f is an L(2,1)-coloring of G}. A no-hole coloring is defined to be an L(2,1)-coloring of a graph with span k which uses all the colors from {0,1,…,k}, for some integer k not necessarily the span of the graph. An L(2,1)-coloring of a graph is said to be irreducible if color of no vertex can be decreased and yield another L(2,1)-coloring of the same graph. An irreducible no-hole coloring of a graph G, in short inh-coloring of G, is an L(2,1)-coloring of G which is both irreducible and no-hole. The lower inh-span or simply inh-span of a graph G, denoted by λinh(G), is defined as λinh(G)=min{span f:f is an inh-coloring of G}. The direct product of graphs G and H is the graph G×H with vertex set V(G)×V(H) in which two vertices (x,y) and (x′,y′) are adjacent if and only if xx′∈E(G) and yy′∈E(H). In this paper we prove that for m≥4,n≥3, Km×Pn is inh-colorable and give the exact value of inh-span of it except in one case, where we give an upper bound of it. We find the exact value of λ(Km×Cn) when m(≥3) is odd and n is a multiple of 5m+1, and give an improved upper bound of λ(Km×Cn) for some other values of m and n. We prove that for m≥3,n≥4, Km×Cn is inh-colorable and give the exact value of λinh(Km×Cn) for n=4,5 or 6 and an upper bound of it for the remaining values of n. We find the value of λ(Pn×Cm) for the cases left by Jha et al. (2005) except the case m=5. For n≥6 and except finitely many values of m we show that Pn×Cm is inh-colorable and find the exact value of λinh(Pn×Cm). We also give two upper bounds for the span of the direct product of two arbitrary graphs.