No-hole L(2, 1)-colorings

نویسندگان

Peter C. Fishburn

Fred S. Roberts

چکیده

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).

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