L(p,q)labeling of sparse graphs
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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.
References
 Calamoneri T (2006) The L(h,k)labeling problem: a survey and an annotated bibliography. Comput J 49:585–608 CrossRefGoogle Scholar
 Chang GJ, Kuo D (1996) The L(2,1)labeling problem on graphs. SIAM J Discrete Math 9:309–316 MathSciNetCrossRefMATHGoogle Scholar
 Charpentier C, Montassier M, Raspaud A (2012) L(p,q)labeling of sparse graphs. Technical report, LaBRI Google Scholar
 Gonçalves D (2005) On the L(p,1)labelling of graphs. Discrete Math 308:1405–1414 CrossRefGoogle Scholar
 Griggs JR, Yeh RK (1992) Labeling graphs with a condition at distance 2. SIAM J Discrete Math 5:586–595 MathSciNetCrossRefMATHGoogle Scholar
 Jensen TR, Toft B (1995) Choosability versus chromaticity. Geocombinatorics 5:45–64 MathSciNetMATHGoogle Scholar
 Kramer F, Kramer H (1969) Un problème de coloration des sommets d’un graphe. C R Acad Sci Paris A 268:46–48 MATHGoogle Scholar
 Kramer F, Kramer H (2008) A survey on the distance coloring of graphs. Discrete Math 308:422–426 MathSciNetCrossRefMATHGoogle Scholar
 Lih KW, Wang WF (2003) Labeling planar graphs with conditions on girth and distance two. SIAM J Discrete Math 17(2):264–275 MathSciNetCrossRefMATHGoogle Scholar
 Lih KW, Wang WF (2006) Coloring the squares of an outerplanar graph. Taiwan J Math 10(4):1015–1023 MathSciNetMATHGoogle Scholar
 Wegner G (1977) Graphs with given diameter and a colouring problem. Technical report, University of Dortmund Google Scholar