The Weak (2, 2)-Labelling Problem for graphs with forbidden induced structures

Julien Bensmail (1) , Hervé Hocquard (2) , Pierre-Marie Marcille (2)

1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

The Weak $(2,2)$-Conjecture is a graph labelling problem asking whether all connected graphs of at least three vertices can have their edges assigned red labels~$1$ and $2$ and blue labels~$1$ and~$2$ so that any two adjacent vertices are distinguished either by their sums of incident red labels, or by their sums of incident blue labels. This problem emerged in a recent work aiming at proposing a general framework encapsulating several distinguishing labelling problems and notions, such as the well-known 1-2-3 Conjecture, a few of its variants, and so-called locally irregular decompositions. One further point of interest behind the Weak $(2,2)$-Conjecture is that it is weaker than the 1-2-3 Conjecture, in the sense that the latter conjecture, if proved true, would imply the former one is true too. In this work, we prove that the Weak $(2,2)$-Conjecture holds for two classes of graphs defined in terms of forbidden induced structures, namely claw-free graphs and graphs with no pair of independent edges. One main point of interest for focusing on such classes of graphs is that the 1-2-3 Conjecture is not known to hold for them. Also, these two classes of graphs have unbounded chromatic number, while the 1-2-3 Conjecture is mostly understood for classes with bounded and low chromatic number.

Mots clés: distinguishing labelling; 1-2-3 Conjecture; sum distinction

Domaines: Mathématique discrète [cs.DM]

Julien Bensmail, Hervé Hocquard, Pierre-Marie Marcille. The Weak (2, 2)-Labelling Problem for graphs with forbidden induced structures. [Research Report] Université côte-d'Azur; Université de Bordeaux, LaBRI, UMR 5800, France. 2022. ⟨hal-03784687v2⟩