On irreducible no-hole L(2, 1)-labelings of hypercubes and triangular lattices

Abstract

An L(2, 1)-labeling (or coloring) of a graph G is a mapping f: V (G) → Z+ (Formula presented){0} such that |f(u) − f(v)| ≥ 2 for all edges uv of G, and |f(u)−f(v)| ≥ 1 if d(u, v) = 2, where d(u, v) is the distance between vertices u and v in G. The span of an L(2, 1)-labeling 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 λ(G), is equal to min {span f: f is an L(2, 1)-labeling of G}. A no-hole labeling (or no-hole coloring) is defined to be an L(2, 1)-labeling with span k which uses all the labels from {0, 1, …, k}, for some integer k not necessarily the span of the graph. An L(2, 1)-labeling is defined as irreducible if no labels of vertices in the graph can be decreased and yield another L(2, 1)-labeling of the same graph. An irreducible no-hole labeling is called an inh-labeling (or inhcoloring). 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-labeling of G}. The upper inh-span of a graph G, denoted by Λinh(G), is defined as Λinh(G) = max{span f: f is an inh-labeling of G}. Villalpando and Laskar [8] have shown that Qn is inh-labelable for very few values of n. The same authors [7] have given a conjecture for the inh-span of infinite triangular lattices and have also given both lower and upper bounds of the same for finite triangular lattices. In this paper we prove that the hypercube Qn is inh-labelable for every n ≥ 4 and find upper bounds of its inh-span and upper inh-span. We find the exact value of the inh-span of all triangular lattices.