Formula omitted]-labeling of planar graphs with small girth

Abstract

A k-L(p,q)-labeling of a graph G is a function ϕ:V(G)→{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 of distance two apart. The L(p,q)-labeling number of G, denoted by λp,q(G) is the least k for which G has a k-L(p,q)-labeling. Let p≥q≥1 be integers and let g be the girth of a graph G. In this paper we prove that (1) if G is a planar graph with g≥5, then λp,q(G)≤(2q−1)Δ+6p+10q−8. (2) Let G be a planar graph with g≥6. There exists an integer Δ0(p,q) such that λp,q(G)≤qΔ+2p+q−2 if Δ≥Δ0(p,q). Our results partially generalize some known results and the second result is best possible for the case p=q=1.