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

• Authors: Wang Weifan;Wang Yiqiao
• Venue: Information Processing Letters
• Year: 2006

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) | v ∈ V(G)} = k.In this paper we prove that, if p + q ≥ 3 and G is a K4-minor free graph with maximum degree Δ, then λ(G; p, q) ≤ 2(2p - 1) + (2q - 1) ⌊3Δ/2⌋ - 2. 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 2 ≤ Δ ≤ 3, or λ(G; 1, 1) ≤ ⌊3Δ/2⌋ if Δ ≥ 4.