On generalized Petersen graphs labeled with a condition at distance two

Abstract

An L(2,1)-labeling of graph G is an integer labeling of the vertices in V( G) such that adjacent vertices receive labels which differ by at least two, and vertices which are distance two apart receive labels which differ by at least one. The λ-number of G is the minimum span taken over all L(2,1)-labelings of G. In this paper, we consider the λ-numbers of generalized Petersen graphs. By introducing the notion of a matched sum of graphs, we show that the λ-number of every generalized Petersen graph is bounded from above by 9. We then show that this bound can be improved to 8 for all generalized Petersen graphs with vertex order >12, and, with the exception of the Petersen graph itself, improved to 7 otherwise.