On irreducible no-hole L(2,1)-coloring of subdivision of graphs

An L(2, 1)-coloring (or labeling) of a graph G is a mapping such that for all edges uv of G, and if u and v are at distance two in G. The span of an L(2, 1)-coloring f, denoted by span f, is the largest integer assigned by f to some vertex of the graph. The span of a graph G, denoted by , is min {span . 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 and there is no vertex v in G such that . A no-hole coloring is defined to be an L(2, 1)-coloring with span k which uses all the colors from , for some integer k not necessarily the span of the graph. An L(2, 1)-coloring is said to be irreducible if colors of no 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, also called 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 , is defined as span f : f is an inh-coloring of G}. Given a graph G and a function h from E(G) to , the h-subdivision of G, denoted by , is the graph obtained from G by replacing each edge uv in G with a path of length h(uv). In this paper we show that is inh-colorable for , , except the case and for at least one edge but not for all. Moreover we find the exact value of in several cases and give upper bounds of the same in the remaining.