Distinguishing adjacent vertices by ordering edges

Abstract

The 1-2-3 Conjecture states that for every graph without isolated edges, there exists an edge-weighting from $\{1,2,3\}$ such that adjacent vertices receive distinct sums of weights on their incident edges. In the sequence variant, adjacent vertices are to be distinguished by the sequences of weights on their incident edges. In this paper, we investigate whether, for every graph without isolated edges and for a fixed proper edge colouring (where colours are interpreted as weights), there exists a global total order of the edges such that the resulting sequences of incident weights distinguish adjacent vertices. For connected graphs, we prove that such an order exists whenever there exist two adjacent vertices that have distinct sets of incident weights. This yields a positive answer for every proper edge colouring of a connected non-regular graph or a connected graph of class two. Moreover, a probabilistic argument gives the same conclusion for connected regular graphs of degree at least six.