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Vertex-colouring edge-weightings with two edge weights

Abstract : An edge-weighting vertex colouring of a graph is an edge-weight assignment such that the accumulated weights at the vertices yields a proper vertex colouring. If such an assignment from a set S exists, we say the graph is S-weight colourable. It is conjectured that every graph with no isolated edge is \1, 2, 3\-weight colourable. We explore the problem of classifying those graphs which are \1, 2\ -weight colourable. We establish that a number of classes of graphs are S -weight colourable for much more general sets S of size 2. In particular, we show that any graph having only cycles of length 0 mod 4 is S -weight colourable for most sets S of size 2. As a consequence, we classify the minimal graphs which are not \1, 2\-weight colourable with respect to subgraph containment. We also demonstrate techniques for constructing graphs which are not \1, 2\-weight colourable.
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Mahdad Khatirinejad, Reza Naserasr, Mike Newman, Ben Seamone, Brett Stevens. Vertex-colouring edge-weightings with two edge weights. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2012, Vol. 14 no. 1 (1), pp.1-20. ⟨hal-00990567⟩

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