Some remarks and results on the Standard (2,2)-Conjecture - Baudon, Olivier; Bensmail, Julien; Vayssieres, Lyn (Wydawnictwa AGH, 2026)
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.
Keywords: 1-2-3 Conjecture; 1-2 Conjecture; proper labelling; labelling