Solutions of Some L(2, 1)-Coloring Related Open Problems

An L(2, 1)-coloring (or labeling) of a graph G is a vertex coloring 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 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)-coloring is the maximum color (or label) assigned by it. The span of a graph G is the smallest integer λ such that there exists an L(2, 1)-coloring of G with span λ. An L(2, 1)-coloring of a graph with span equal to the span of the graph is called a span coloring. For an L(2, 1)-coloring f of a graph G with span k, an integer h is a hole in f if h ∈ (0, k) and there is no vertex v in G such that f(v) = h. A no-hole coloring is an L(2, 1)-coloring with no hole in it. An L(2, 1)-coloring is irreducible if color of none of the vertices in the graph can be decreased to yield another L(2, 1)-coloring of the same graph. A graph G is inh-colorable if there exists an irreducible no-hole coloring of G. Most of the results obtained in this paper are answers to some problems asked by Laskar et al. [5]. These problems are mainly about relationship between the span and maximum no-hole span of a graph, lower inh-span and upper inh-span of a graph, and the maximum number of holes and minimum number of holes in a span coloring of a graph. We also give some sufficient conditions for a tree and an unicyclic graph to have inh-span Δ + 1.