On (d,1)-total numbers of graphs

A (d,1)(d,1)-total labelling of a graph GG assigns integers to the vertices and edges of GG 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 dd. The span of a (d,1)(d,1)-total labelling is the maximum difference between two labels. The (d,1)(d,1)-total number, denoted λdT(G), is defined to be the least span among all (d,1)(d,1)-total labellings of GG. We prove new upper bounds for λdT(G), compute some λdT(Km,n) for complete bipartite graphs Km,nKm,n, and completely determine all λdT(Km,n) for d=1,2,3d=1,2,3. We also propose a conjecture on an upper bound for λdT(G) in terms of the chromatic number and the chromatic index of GG.