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 with integers, where a vertex colour is simply a sum of the weight assigned to the vertex itself and the weights of its incident edges. Can we obtain a proper colouring using only weights 1 and 2 for an arbitrary G? We give a positive answer when G is a 3-colourable, complete or 4-regular graph. We also show that it is enough to C use weights from 1 to 11, as well as from 1 to 11 [chi(G)/2] + 1, for an arbitrary graph G.

Source : oai:HAL:hal-00990444v1

Volume: Vol. 12 no. 1

Section: Graph and Algorithms

Published on: January 1, 2010

Submitted on: March 26, 2015

Keywords: neighbour-distinguishing total-weighting,irregularity strength,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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