@MISC{Havet12griggsand, author = {F. Havet and et al.}, title = {Griggs and Yeh's conjecture . . .}, year = {2012} }

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Abstract

An L(p, 1)-labeling of a graph is a function f from the vertex set to the positive integers such that |f(x) − f(y) | p if dist(x, y) = 1 and |f(x) − f(y) | 1 if dist(x, y) = 2, where dist(x, y) is the distance between the two vertices x and y in the graph. The span of an L(p, 1)-labeling f is the difference between the largest and the smallest labels used by f. In 1992, Griggs and Yeh conjectured that every graph with maximum degree Δ 2 has an L(2, 1)-labeling with span at most Δ2. We settle this conjecture for Δ sufficiently large. More generally, we show that for any positive integer p there exists a constant Δp such that every graph with maximum degree Δ Δp has an L(p, 1)-labeling with span at most Δ2. This yields that for each positive integer p, there is an integer Cp such that every graph with maximum degree Δ has an L(p, 1)-labeling with span at most Δ2 + Cp.