On irreducible no-hole L(2, 1)-coloring of Cartesian product of trees with paths

Abstract

An L(2, 1)-coloring of a graph G is a mapping f : VðGÞ ! Z þ [ f0g such that jf ðuÞ À f ðvÞj ! 2 for all edges uv of G, and jf ðuÞ À f ðvÞj ! 1 if u and v are at distance two in G. The span of an L(2, 1)coloring f of G, denoted by span(f), is max ff ðvÞ : v 2 VðGÞg: The span of G, denoted by kðGÞ, is the minimum span of all possible L(2, 1)-colorings of G. If f is an L(2, 1)-coloring of a graph G with span k then an integer l is a hole in f if l 2 ð0, kÞ and there is no vertex v in G such that f(v) ¼ l. A no-hole coloring is defined to be an L(2, 1)-coloring with no hole in it. An L(2, 1)-coloring is said to be irreducible if the color of none of the vertices in the graph 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. A graph G is inh-colorable if there exists an inh-coloring of it. For an inh-colorable graph G the lower inh-span or simply inh-span of G, denoted by k inh ðGÞ, is defined as k inh ðGÞ ¼ minfspan ðf Þ : f is an inh-coloring of G}. In this paper, we prove that the Cartesian product of trees with paths are inh-colorable. KEYWORD L(2, 1)-coloring; no-hole coloring; irreducible coloring; span of a graph; Cartesian product of graphs 2010 MSC