A Note on Neighbour-Distinguishing Regular Graphs Total-Weighting

We investigate the following modification of a problem posed by Karoński, Luczak and Thomason [J. Combin. Theory, Ser. B 91 (2004) 151-157]. Let us assign positive integers to the edges and vertices of a simple graph G. As a result we obtain a vertex-colouring of G by sums of weights assigned to the vertex and its adjacent edges. Can we obtain a proper colouring using only weights 1 and 2 for an arbitrary G? We know that the answer is yes if G is a 3-colourable, complete or 4-regular graph. Moreover, it is enough to use weights from 1 to 11, as well as from 1 to b 2 c + 1, for an arbitrary graph G. Here we show that weights from 1 to 7 are enough for all regular graphs.