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Journal Articles SIAM Journal on Discrete Mathematics Year : 2012

Griggs and Yeh's Conjecture and L(p,1)-labelings

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 Δ^2. We settle this conjecture for Δ sufficiently large. More generally, we show that for any positive integer p there exists a constant Δ_p such that every graph with maximum degree Δ ≥ Δ_p has an L(p,1)-labeling with span at most Δ^2. This yields that for each positive integer p, there is an integer C_p such that every graph with maximum degree Δ has an L(p,1)-labeling with span at most Δ^2 + C_p.
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Dates and versions

inria-00327909 , version 1 (09-10-2008)
inria-00327909 , version 2 (05-03-2012)

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Frédéric Havet, Bruce Reed, Jean-Sébastien Sereni. Griggs and Yeh's Conjecture and L(p,1)-labelings. SIAM Journal on Discrete Mathematics, 2012, 26 (1), pp.145--168. ⟨10.1137/090763998⟩. ⟨inria-00327909v2⟩
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