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arXiv · 2504.21452

1-2 Conjectures for Graphs with Low Degeneracy Properties

Julien Bensmail·Beatriz Martins·Chaoliang Tang
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Abstract

In a recent work, Keusch proved the so-called 1-2-3 Conjecture, raised by Karo\'nski, {\L}uczak, and Thomason in 2004: for every connected graph different from $K_2$, we can assign labels~$1,2,3$ to the edges so that no two adjacent vertices are incident to the same sum of labels. Despite this significant result, several problems close to the 1-2-3 Conjecture in spirit remain widely open. In this work, we focus on the so-called 1-2 Conjecture, raised by Przyby{\l}o and Wo\'zniak in 2010, which is a counterpart of the 1-2-3 Conjecture where labels~$1,2$ only can be assigned, and both vertices and edges are labelled. We consider both the 1-2 Conjecture in its original form, where adjacent vertices must be distinguished w.r.t.~their sums of incident labels, and variants for products and multisets. We prove some of these conjectures for graphs with bounded maximum degree (at most~$6$) and bounded maximum average degree (at most~$3$), going beyond earlier results of the same sort.

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Julien Bensmail, Beatriz Martins, Chaoliang Tang. 2025-04-30. 1-2 Conjectures for Graphs with Low Degeneracy Properties. https://arxiv.org/abs/2504.21452

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