On 1-2-3 Conjecture-like problems in 2-edge-coloured graphs

Abstract

The well-known 1-2-3 Conjecture asks whether almost all graphs can have their edges labelled with $1,2,3$ so that any two adjacent vertices are distinguished w.r.t. the sums of their incident labels. This conjecture has attracted increasing attention over the last years, with many of its aspects of interest being investigated by several authors. In early 2023, Keusch proposed a full solution to the 1-2-3 Conjecture.

Among other aspects of interest, several works introduced and studied ways of generalising such distinguishing labellings and the 1-2-3 Conjecture to structures more general than graphs, such as digraphs and hypergraphs. In the current work, we introduce two new variants for 2-edge-coloured graphs (having negative and positive edges), in which, through labellings, pairs of adjacent vertices are considered distinguished if and only if the differences between their incident positive and negative sums are different. The difference between the two variants we introduce is that, in one of them, this distinction must be met even when considering the absolute value of these differences.

We investigate how these two variants connect, and how they relate to the original problem. For each of the two variants, we also establish upper bounds on the minimum number of consecutive labels that suffice to design a distinguishing labelling of almost any 2-edge-coloured graph. This leads us to raise some conjectures on this minimum, which, as support, we prove for some families of 2-edge-coloured graphs. We also investigate weaker versions of these conjectures, where one can choose the polarity of the edges.