Vertex-colouring edge-weightings with two edge weights.

*(English)*Zbl 1283.05105Summary: 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.