Edge colouring by total labellings

We introduce the concept of an edge-colouring total k -labelling. This is a labelling of the vertices and the edges of a graph G with labels 1 , 2 , … , k such that the weights of the edges define a proper edge colouring of G . Here the weight of an edge is the sum of its label and the labels of its two endvertices. We define χ t ′ ( G ) to be the smallest integer k for which G has an edge-colouring total k -labelling. This parameter has natural upper and lower bounds in terms of the maximum degree Δ of G : ⌈ ( Δ + 1 ) / 2 ⌉ ≤ χ t ′ ( G ) ≤ Δ + 1 . We improve the upper bound by 1 for every graph and prove χ t ′ ( G ) ≤ Δ / 2 + O ( Δ log Δ ) . Moreover, we investigate some special classes of graphs.