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 λTd (G), is defined to be the least span among all (d, 1)-total labellings of G. We prove new upper bounds for λTd (G), compute some λ T d (Km,n) for complete bipartite graphsKm,n, and completely determine all λd (Km,n) for d = 1, 2, 3. We also propose a conjecture on an upper bound for λTd (G) in terms of the chromatic number and the chromatic index of G.