On Local Irregularity Conjecture for 2-multigraphs

TL;DR

This paper proves the Local Irregularity Conjecture for 2-multigraphs in specific graph classes and establishes bounds for the chromatic index in planar and subcubic graphs, advancing understanding of irregular edge colorings.

Contribution

The paper confirms the conjecture for regular, split, and certain subcubic graphs, and provides bounds for planar and subcubic 2-multigraphs, connecting to the 1-2-3 Conjecture.

Findings

Confirmed the conjecture for regular graphs

Established bounds for planar 2-multigraphs

Improved bounds for subcubic graphs

Abstract

A multigraph in which adjacent vertices have different degrees is called locally irregular. The locally irregular edge coloring is an edge coloring of a multigraph $G$ in which every color induces a locally irregular submultigraph of $G$. We denote by $\operatorname{lir}(G)$ the locally irregular chromatic index of a multigraph $G$, which is the smallest number of colors required in a locally irregular edge coloring of $G$, given that such a coloring of $G$ exists. By $^2G$ we denote a 2-multigraph obtained from a simple graph $G$ by doubling each its edge. In 2022 Grzelec and Wo\'zniak conjectured that $\operatorname{lir}(^2G) \leq 2$ for every connected simple graph $G$ different from $K_2$; the conjecture is known as Local Irregularity Conjecture for 2-multigraphs. In this paper, we prove this conjecture in the case of regular graphs, split graphs, and some particular families of subcubic graphs. Moreover, we provide a constant upper bound on the locally irregular chromatic index of planar 2-multigraphs (except for $^2K_2$), and we obtain a better constant upper bound on $\operatorname{lir}(^2G)$ if $G$ is a simple subcubic graph different from $K_2$. In the proofs, special decompositions of graphs and the relation of Local Irregularity Conjecture to the well-known 1-2-3 Conjecture are utilized.