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 χt(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 ∆ of G: ⌈(∆ + 1)/2⌉ ≤ χt(G) ≤ ∆ + 1. We improve the upper bound by 1 for every graph and prove a general upper bound of χt(G) ≤ ∆/2 +O( √ ∆ log∆). Moreover, we investigate some special classes of graphs.