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**11 - 12**of**12**### Griggs and Yeh's conjecture . . .

, 2012

"... 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 ..."

Abstract
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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.

### Distance constrained labelings of planar graphs with no short cycles

, 2007

"... Motivated by a conjecture of Wang and Lih, we show that every planar graph of girth at least 7 and maximum degree ∆ ≥ 190 + 2⌈p/q ⌉ has an L(p, q)-labeling of span at most 2p+q∆−2. Since the optimal span of an L(p, 1)-labeling of an infinite ∆-regular tree is 2p+∆ − 2, the obtained bound is the bes ..."

Abstract
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Motivated by a conjecture of Wang and Lih, we show that every planar graph of girth at least 7 and maximum degree ∆ ≥ 190 + 2⌈p/q ⌉ has an L(p, q)-labeling of span at most 2p+q∆−2. Since the optimal span of an L(p, 1)-labeling of an infinite ∆-regular tree is 2p+∆ − 2, the obtained bound is the best possible for any p ≥ 1 and q = 1.