Group path covering and L(j, k)-labelings of diameter two graphs

L(j,k)-labeling is a kind of generalization of the classical graph coloring motivated from a kind of frequency assignment problem in radio networks, in which adjacent vertices are assigned integers that are at least j apart, while vertices that are at distance two are assigned integers that are at least k apart. The span of an L(j. k)-labeling of a graph G is the difference between the maximum and the minimum integers assigned to its vertices. The L(j, k)-labeling number of G, denoted by lambda(j,k)(G), is the minimum span over all L( j, k)labelings of G. Georges, Mauro and Whittlesey (1994) [1] established the relationship between lambda(2,1) (G) of a graph G and the path covering number of GC (the complement of G). Georges, Mauro and Stein (2000) [2] determined the L( j. k)-labeling numbers of Cartesian products of two complete graphs. Lam, Lin and Wu (2007) [3] determined the lambda(j,k)-numbers of direct products of two complete graphs. In 2011, we (Wang and Lin, 2011 [4]) generalized the concept of the path covering to the t-group path covering of a graph where t (>= 1) is an integer and established the relationship between the L'(d, 1)-labeling number (d >= 2) of a graph G and the (d - 1)-group path covering number of Cc. In this paper, we establish the relationship between the lambda(j.k)(G) of a graph G with diameter 2 and the left perpendicularj/kright perpendicular-group path coverings of G(c). Using those results, we can have shorter proofs to obtain the X j,k of the Cartesian products and direct products of complete graphs. (C) 2011 Elsevier B.V. All rights reserved.