Formula omitted]-labelling of [formula omitted]-minor free graphs

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 ) | v ∈ V ( G ) } = k . In this paper we prove that, if p + q ⩾ 3 and G is a K 4 -minor free graph with maximum degree Δ, then λ ( G ; p , q ) ⩽ 2 ( 2 p − 1 ) + ( 2 q − 1 ) ⌊ 3 Δ / 2 ⌋ − 2 . This generalizes a result by Lih et al. [K.W. Lih, W.F. Wang, X. Zhu, Coloring the square of a K 4 -minor free graph, Discrete Math. 269 (2003) 303–309], which says that every K 4 -minor free graph G has λ ( G ; 1 , 1 ) ⩽ Δ + 2 if 2 ⩽ Δ ⩽ 3 , or λ ( G ; 1 , 1 ) ⩽ ⌊ 3 Δ / 2 ⌋ if Δ ⩾ 4 .