On (d,1)-total numbers of graphs

A ( d , 1 ) -total labelling of a graph G assigns integers to the vertices and edges of G such that adjacent vertices receive distinct labels, adjacent edges receive distinct labels, and a vertex and its incident edges receive labels that differ in absolute value by at least d . The span of a ( d , 1 ) -total labelling is the maximum difference between two labels. The ( d , 1 ) -total number, denoted λ d T ( G ) , is defined to be the least span among all ( d , 1 ) -total labellings of G . We prove new upper bounds for λ d T ( G ) , compute some λ d T ( K m , n ) for complete bipartite graphs K m , n , and completely determine all λ d T ( K m , n ) for d = 1 , 2 , 3 . We also propose a conjecture on an upper bound for λ d T ( G ) in terms of the chromatic number and the chromatic index of G .