Group path covering and distance two labeling of graphs

Abstract

For a positive integer d, an L ( d , 1 ) -labeling f of a graph G is an assignment of integers to the vertices of G such that | f ( u ) − f ( v ) | ⩾ d if u v ∈ E ( G ) , and | f ( u ) − f ( v ) | ⩾ 1 if u and u are at distance two. The span of an L ( d , 1 ) -labeling f of a graph is the absolute difference between the maximum and minimum integers used by f. The L ( d , 1 ) -labeling number of G, denoted by λ d , 1 ( G ) , is the minimum span over all L ( d , 1 ) -labelings of G. An L ′ ( d , 1 ) -labeling of a graph G is an L ( d , 1 ) -labeling of G which assigns different labels to different vertices. Denote by λ d , 1 ′ ( G ) the L ′ ( d , 1 ) -labeling number of G. Georges et al. [Discrete Math. 135 (1994) 103–111] established relationship between the L ( 2 , 1 ) -labeling number of a graph G and the path covering number of G c , the complement of G. In this paper we first generalize the concept of the path covering of a graph to the t-group path covering. Then we establish the relationship between the L ′ ( d , 1 ) -labeling number of a graph G and the ( d − 1 ) -group path covering number of G c . Using this result, we prove that λ 2 , 1 ′ ( G ) and λ 3 , 1 ′ ( G ) for bipartite graphs G can be computed in polynomial time.