On a combination of the 1-2-3 Conjecture and the Antimagic Labelling Conjecture

TL;DR

This paper investigates whether edges of any graph can be weighted uniquely to distinguish adjacent vertices, combining the 1-2-3 Conjecture and the Antimagic Labelling Conjecture, and provides positive results for various graph classes.

Contribution

It proves that many graph classes, including trees and regular graphs, admit such weightings, and introduces methods to extend these results to general graphs with additional weights.

Findings

Regular graphs admit such weightings.

Trees admit such weightings, confirming a recent conjecture.

Graphs with maximum average degree 3 require only a small number of extra weights.

Abstract

This paper is dedicated to studying the following question: Is it always possible to injectively assign the weights $1,...,|E(G)|$ to the edges of any given graph $G$ (with no component isomorphic to $K_2$) so that every two adjacent vertices of $G$ get distinguished by their sums of incident weights? One may see this question as a combination of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture. Throughout this paper, we exhibit evidence that this question might be true. Benefiting from the investigations on the Antimagic Labelling Conjecture, we first point out that several classes of graphs, such as regular graphs, indeed admit such assignments. We then show that trees also do, answering a recent conjecture of Arumugam, Premalatha, Ba\v{c}a and Semani\v{c}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a. Towards a general answer to the question above, we then prove that claimed assignments can be constructed for any graph, provided we are allowed to use some number of additional edge weights. For some classes of sparse graphs, namely $2$-degenerate graphs and graphs with maximum average degree~$3$, we show that only a small (constant) number of such additional weights suffices.