Making Graphs Irregular through Irregularising Walks

TL;DR

This paper explores the problem of transforming graphs into locally irregular multigraphs using walks, focusing on the shortest such walks and their properties under various constraints.

Contribution

It introduces a new constrained version of the irregularising problem, analyzing the impact of walk-based edge additions on graph irregularity.

Findings

Characterization of shortest irregularising walks

Structural and combinatorial bounds established

Algorithmic approaches for specific graph classes

Abstract

The 1-2-3 Conjecture, introduced by Karo\'nski, {\L}uczak, and Thomason in 2004, was recently solved by Keusch. This implies that, for any connected graph $G$ different from $K_2$, we can turn $G$ into a locally irregular multigraph $M(G)$, i.e., in which no two adjacent vertices have the same degree, by replacing some of its edges with at most three parallel edges. In this work, we introduce and study a restriction of this problem under the additional constraint that edges added to $G$ to reach $M(G)$ must form a walk (i.e., a path with possibly repeated edges and vertices) of $G$. We investigate the general consequences of having this additional constraint, and provide several results of different natures (structural, combinatorial, algorithmic) on the length of the shortest irregularising walks, for general graphs and more restricted classes.