No-hole L(2,1)-colorings

An L(2, 1)-coloring of a graph G is a coloring of G's vertices with integers in {0, 1,...k} so that adjacent vertices' colors differ by at least two and colors of distance-two vertices differ. We refer to an L(2, 1)-coloring as a coloring. The span lambda(G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is lambda(G), and the hole index rho(G) of G is the minimum number of colors in {0,1,...,lambda(G)} not used in a span coloring. We say that G is full-colorable if rho(G) = 0. More generally, a coloring of G is a no-hole coloring if it uses all colors between 0 and its maximum color. Both colorings and no-hole colorings were motivated by channel assignment problems. We define the no-hole span mu(G) of G as 00 if G has no no-hole coloring, otherwise mu(G) is the minimum k for which G has a no-hole coloring using colors in {0, 1,...k}. Let n denote the number of vertices of G, and let Delta be the maximum degree of vertices of G. Prior work shows that all non-star trees with Delta greater than or equal to 3 are full-colorable, all graphs G with n=lambda(G) + 1 are full-colorable, mu(G) less than or equal to lambda(G) + rho(G) if G is not full-colorable and n greater than or equal to lambda(G) + 2, and G has a no-hole coloring if and only if n greater than or equal to lambda(G) + 1. We prove two extremal results for colorings. First, for every m greater than or equal to 1 there is a G with rho(G) = m and mu(G) = lambda(G) + m. Second, for every m greater than or equal to 2 there is a connected G with lambda(G) = 2m, n = lambda(G) + 2 and rho(G) = m. (C) 2003 Elsevier B.V. All rights reserved.