L(p,q)‐labeling of sparse graphs

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