L(2, 1)-Labeling of Permutation and Bipartite Permutation Graphs

Abstract

An L(2, 1)-labeling of a graph is an assignment of non-negative integers, called colours to the vertex set of G such that the difference between the colours assigned to adjacent vertices is at least two and the colours assigned to any two vertices at distance two are distinct. The L(2, 1)-labeling number λ2,1(G) of G is the minimum range of label over all such possible labelings. It was shown by Bodlaender et al. (Comput J 47(2):193–204, 2004) that \({\lambda_{2,1}(G)\leq 5\Delta-2}\) , when G is a permutation graph. In this paper, the authors improve the upper bound for permutation graphs to max\({\{4\Delta-2, 5\Delta-8\}}\) , by doing a detailed analysis of Chang and Kuo’s heuristic for L(2, 1)-labeling of general graphs applied to the particular case of permutation graphs. On the other hand, Araki (Discrete Appl Math 157:1677–1686, 2009) showed that, for a bipartite permutation graph G, \({bc(G)\leq \lambda_{2,1}(G)\leq bc(G)+1}\) , where bc(G) is the biclique number of G. This paper also provides sufficient conditions of bipartite permutation graphs G such that \({\lambda_{2,1}(G)=bc(G)+1}\).