On a List Variant of the Multiplicative 1-2-3 Conjecture

TL;DR

This paper investigates the list version of the Multiplicative 1-2-3 Conjecture, aiming to determine the minimum list size needed for edge labelling to ensure adjacent vertices have different incident products, with results specific to certain graph classes.

Contribution

It introduces the first study of the list variant of the Multiplicative 1-2-3 Conjecture and provides upper bounds on the list size for various classes of graphs.

Findings

Established upper bounds on list size for specific graph classes.

Linked the problem to the List 1-2-3 Conjecture to derive bounds.

Improved some bounds through specialized arguments.

Abstract

The 1-2-3 Conjecture asks whether almost all graphs can be (edge-)labelled with $1,2,3$ so that no two adjacent vertices are incident to the same sum of labels. In the last decades, several aspects of this problem have been studied in literature, including more general versions and slight variations. Notable such variations include the List 1-2-3 Conjecture variant, in which edges must be assigned labels from dedicated lists of three labels, and the Multiplicative 1-2-3 Conjecture variant, in which labels~$1,2,3$ must be assigned to the edges so that adjacent vertices are incident to different products of labels. Several results obtained towards these two variants led to observe some behaviours that are distant from those of the original conjecture. In this work, we consider the list version of the Multiplicative 1-2-3 Conjecture, proposing the first study dedicated to this very problem. In particular, given any graph $G$, we wonder about the minimum~$k$ such that $G$ can be labelled as desired when its edges must be assigned labels from dedicated lists of size~$k$. Exploiting a relationship between our problem and the List 1-2-3 Conjecture, we provide upper bounds on~$k$ when $G$ belongs to particular classes of graphs. We further improve some of these bounds through dedicated arguments.