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 di er by at least two and colors of distance-two vertices di er. We refer to an L(2; 1)-coloring as a coloring. The span (G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is (G), and the hole index (G) of G is the minimum number of colors in {0; 1; : : : ; (G)} not used in a span coloring. We say that G is full-colorable if (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 deΓΏne the no-hole span (G) of G as β if G has no no-hole coloring; otherwise (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 be the maximum degree of vertices of G. Prior work shows that all non-star trees with ΒΏ 3 are full-colorable, all graphs G with n = (G) + 1 are full-colorable, (G) 6 (G) + (G) if G is not full-colorable and n ΒΏ (G) + 2, and G has a no-hole coloring if and only if n ΒΏ (G) + 1. We prove two extremal results for colorings. First, for every m ΒΏ 1 there is a G with (G) = m and (G) = (G) + m. Second, for every m ΒΏ 2 there is a connected G with (G) = 2m, n = (G) + 2 and (G) = m.