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Vertex-colouring edge-weightings with two edge weights. (English) Zbl 1283.05105
Summary: 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 $$0\bmod 4$$ 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.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C22 Signed and weighted graphs
##### Keywords:
edge weighting; graph colouring
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