Vertex-colouring edge-weightings with two edge weights

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.