Holes in L(2, 1)-Coloring on Certain Classes of Graphs

Abstract

The channel assignment problem is the problem of efficiently assigning frequencies to radio transmitters located at various places such that communications do not interfere. Griggs and Yeh [5] introduced a variation of the channel assignment problem known as the L(2, 1)-colorings of graphs. An L(2, 1)-coloring of a graph G = (V, E) is a vertex coloring f: V(G) → {0, 1, 2,…, k}, k ≥ 2 such that |f(u) – f(v)| ≥ 2 for all uv ∊ E(G) and |f(u) – f(v)| ≥ 1 if d(u, v) = 2. The span of G, λ(G), is the smallest integer k for which G has an L(2, 1)-coloring. An L(2, 1)-coloring f is irreducible if there does not exist an L(2, 1)-coloring g such that g(u) ≤ f(u) for all u ∊ V(G) and g(v) < f(v) for some v ∊ V(G). A span coloring is an L(2, 1)-coloring whose greatest color is λ(G). Let f be an L(2, 1)-coloring that uses colors from 0 to k. Then h ∊ (0, k) is a hole if there is no vertex v in V(G) such that f(v) = h. In this paper, we investigate maximum number of holes in span colorings of certain classes of graphs. We give exact values for the maximum number of holes in a span coloring of a path, cycle, star, complete bipartite graph and characterize complete graphs in terms of their maximum number of holes. Upper bounds for an arbitrary graph and other classes of graphs are also given.