Journal of Combinatorial Optimization

, Volume 25, Issue 4, pp 646–660

# L(p,q)-labeling of sparse graphs

Article

## 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).

## Keywords

Graph theory Graph labeling Graph coloring Sparse graphs

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## Authors and Affiliations

• Clément Charpentier
• 1
• Mickaël Montassier
• 1
• André Raspaud
• 1
Email author
1. 1.LaBRIUniversité de BordeauxTalence cedexFrance