On a 1, 2 Conjecture

Abstract : 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.
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Jakub Przybylo, Mariusz Woźniak. On a 1, 2 Conjecture. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2010, 12 (1), pp.101-108. ⟨hal-00990444⟩

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