L(p,q)-labelling of K4-minor free graphs

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Abstract

For positive integers p and q, an L(p,q)-labelling of a graph G is a function φ from the vertex set V(G) to the integer set {0,1,,k} such that |φ(x)φ(y)|p if x and y are adjacent and |φ(x)φ(y)|q if x and y are at distance 2. The L(p,q)-labelling number λ(G;p,q) of G is the smallest k such that G has an L(p,q)-labelling with max{ϕ(v)|vV(G)}=k.

In this paper we prove that, if p+q3 and G is a K4-minor free graph with maximum degree Δ, then λ(G;p,q)2(2p1)+(2q1)3Δ/22. This generalizes a result by Lih et al. [K.W. Lih, W.F. Wang, X. Zhu, Coloring the square of a K4-minor free graph, Discrete Math. 269 (2003) 303–309], which says that every K4-minor free graph G has λ(G;1,1)Δ+2 if 2Δ3, or λ(G;1,1)3Δ/2 if Δ4.

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Research supported partially by NSFC (No. 10471131) and ZJNSF (No. M103094, Y604167).

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