L(p,q)-labeling of sparse graphs

Abstract

Let p and q be positive integers. An L(p,q)-labeling of a graph G with a span s is a labeling of its vertices by integers between 0 and s such that adjacent vertices of G are labeled using colors at least p apart, and vertices having a common neighbor are labeled using colors at least q apart. We denote by λ _p,q (G) the least integer k such that G has an L(p,q)-labeling with span k.

The maximum average degree of a graph G, denoted by $\operatorname {Mad}(G)$, is the maximum among the average degrees of its subgraphs (i.e. $\operatorname {Mad}(G) = \max\{\frac{2|E(H)|}{|V(H)|} ; H \subseteq G \}$). We consider graphs G with $\operatorname {Mad}(G) < \frac{10}{3}$, 3 and $\frac{14}{5}$. These sets of graphs contain planar graphs with girth 5, 6 and 7 respectively.

We prove in this paper that every graph G with maximum average degree m and maximum degree Δ has:

• λ _p,q (G)≤(2q−1)Δ+6p+10q−8 if $m < \frac{10}{3}$ and p≥2q.

• λ _p,q (G)≤(2q−1)Δ+4p+14q−9 if $m < \frac{10}{3}$ and 2q>p.

• λ _p,q (G)≤(2q−1)Δ+4p+6q−5 if m<3.

• λ _p,q (G)≤(2q−1)Δ+4p+4q−4 if $m < \frac{14}{5}$.

We give also some refined bounds for specific values of p, q, or Δ. By the way we improve results of Lih and Wang (SIAM J. Discrete Math. 17(2):264–275, 2003).