Group path covering and [formula omitted]-labelings of diameter two graphs

Abstract

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 λ 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 λ 2 , 1 ( G ) of a graph G and the path covering number of G c (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 λ 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 G c . In this paper, we establish the relationship between the λ j , k ( G ) of a graph G with diameter 2 and the ⌊ j / k ⌋ -group path coverings of G c . Using those results, we can have shorter proofs to obtain the λ j , k of the Cartesian products and direct products of complete graphs.