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. \(\operatorname {Mad}(G) = \max\{\frac{2E(H)}{V(H)} ; H \subseteq G \}\)). 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.

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 graphsNotes
Acknowledgements
The authors are grateful to the referees for their careful reading and their helpful comments.
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