Labelling of some planar graphs with a condition at distance two

Abstract

The problem of vertex labeling with a condition at distance two in a graph, is a variation of Hale's channel assignment problem, which was first explored by Griggs and Yeh. For positive integer p ≥ q, the λp,q-number of graph G, denoted λ(G;p, q), is the smallest span among all integer labellings of V(G) such that vertices at distance two receive labels which differ by at least q and adjacent vertices receive labels which differ by at least p. Van den Heuvel and McGuinness have proved that λ(G;p, q) ≤ (4q-2)Δ+10p+38q-24 for any planar graph G with maximum degree Δ. In this paper, we studied the upper bound of λp,q-number of some planar graphs. It is proved that λ(G;p,q) ≤ (2q-1)Δ+2(2p-1) if G is an outerplanar graph and λ(G;p, q) ≤ (2q-1)Δ+6p-4q-1 if G is a Halin graph.