1-2 Conjectures for Graphs with Low Degeneracy Properties

TL;DR

The paper investigates variants of the $1$-$2$ Conjecture, focusing on total, multiset, and product-distinguishing labelings with labels from $\\{1,2\\}$. It presents constructive proofs showing that: (i) for graphs with maximum degree $\Delta(G)\le 6$, there exists a $2$-label total labelling that is product-proper, (ii) for graphs with maximum average degree ${\rm mad}(G)\le 3$, the multiset variant admits a $2$-label total labelling that is multiset-proper, and (iii) as corollaries, the sum-distinguishing variant holds for $4$-, $5$-, and $6$-regular graphs. The methods are constructive and yield polynomial-time algorithms, with discharging used to handle the ${\rm mad}\le 3$ case. Together, these results extend the classes of graphs for which the various $1$-$2$ conjecture variants are known to hold, and provide a framework for further exploration of higher degrees and other variants (sum/product/multiset) in a rigorous, algorithmic fashion.

Abstract

In a recent work, Keusch proved the so-called 1-2-3 Conjecture, raised by Karoński, Łuczak, 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 Przybyło and Woźniak 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.

Figures (5)

• Figure 1: Representation of $G$ as $X\cup Y\cup R$. The matching $M$ is represented with red edges.
• Figure 2: Illustration of the structure of $R$, in the proof of Theorem \ref{['theorem:max-degree-5']}.
• Figure 3: Vertices of $T_1$, $T_2$, and $T_4$ in the proof of Theorem \ref{['theorem:max-degree-6']}. Vertices highlighted in red are faulty vertices. Vertices highlighted in green are part of $I$, and thus incident to an edge of $M$. Dashed edges are edges that may or may not be in $R$.
• Figure 4: Vertices of $T_3$ and $T_5$ in the proof of Theorem \ref{['theorem:max-degree-6']}. Vertices highlighted in red are faulty vertices. Vertices highlighted in green are part of $I$, and thus incident to an edge of $M$.
• Figure 5: Reducible configurations in the proof of Theorem \ref{['theorem:mad3']}

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Hall's Marriage Theorem Hal35
• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Lemma 4.2