1 Permanent Member.pdfVIP

DIMACS Technical Report 2001-10 March 2001 No-Hole L(2,1)-Colorings by Peter C. Fishburn1 ATT Shannon Laboratory Florham Park, NJ 07932 Fred S. Roberts2 Rutgers University Piscataway, NJ 08854 1Permanent Member 2Permanent Member and Director DIMACS is a partnership of Rutgers University, Princeton University, ATT Labs-Research, Bell Labs, NEC Research Institute and Telcordia Technologies (formerly Bellcore). DIMACS is an NSF Science and Technology Center, funded under contract STC–91–19999; and also receives support from the New Jersey Commission on Science and Technology. ABSTRACT 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 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}. We prove that G is full-colorable if it has λ(G)+1 vertices. In addition, if G is not full-colorable and if it has at least λ(G)+2 vertices, then μ(G) ≤ λ(G) + ρ(G). Moreover, for each m ≥ 1 there is a graph with ρ(G) = m and μ(G) = λ(G) + ρ(G). 1 INTRODUCTION Let G be the family of simple graphs with nonempty, finite vertex sets. A coloring of G = (V,E) in G is a map f : V → Z. It is an ordinary coloring if f(x) 6= f(y) whenever {x, y} ∈ E, and a no-hole coloring if f(V ) is a set of consecutive integers. A