Some remarks and results on the Standard (2,2)-Conjecture

Abstract

In this note, we prove that every graph can be edge-labelled with red labels (1,2) and blue labels (1,2) so that vertices with any sum of incident red labels induce a (1)-degenerate graph, while vertices with any sum of incident blue labels induce a (0)-degenerate graph. This result stands as a closer step towards the so-called Standard ((2,2))-Conjecture (stating that (0)-degeneracy can be achieved in both colours), and provides some insight on the surrounding field, which covers the 1-2-3 Conjecture, the 1-2 Conjecture, and other close problems. Along the way, we also describe how many related problems are interconnected, and raise new problems and questions for further work on the topic.