No-hole L(2,1)-colorings

An L(2,1)-coloring of a graph G is a coloring of Gʹs 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 λ(G) of G is the smallest k for which G has a coloring, a span coloring is a coloring whose greatest color is λ(G), and the hole index ρ(G) of G is the minimum number of colors in {0,1,…,λ(G)} not used in a span coloring. We say that G is full-colorable if ρ(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 μ(G) of G as ∞ if G has no no-hole coloring; otherwise μ(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 Δ be the maximum degree of vertices of G. Prior work shows that all non-star trees with Δ⩾3 are full-colorable, all graphs G with n=λ(G)+1 are full-colorable, μ(G)⩽λ(G)+ρ(G) if G is not full-colorable and n⩾λ(G)+2, and G has a no-hole coloring if and only if n⩾λ(G)+1. We prove two extremal results for colorings. First, for every m⩾1 there is a G with ρ(G)=m and μ(G)=λ(G)+m. Second, for every m⩾2 there is a connected G with λ(G)=2m, n=λ(G)+2 and ρ(G)=m.