On decomposing multigraphs into locally irregular submultigraphs

TL;DR

This paper investigates how to decompose multigraphs into subgraphs where adjacent vertices have different degrees, proposing a conjecture and proving it for various graph classes, advancing understanding of locally irregular edge colorings.

Contribution

It introduces a conjecture relating to the locally irregular chromatic index of doubled graphs and proves it for several important graph classes.

Findings

Conjecture holds for paths, cycles, wheels, complete graphs, and bipartite graphs.

Established bounds for the locally irregular chromatic index of 2-multigraphs.

Connected to well-known conjectures like the 1-2-3 Conjecture.

Abstract

A locally irregular multigraph is a multigraph whose adjacent vertices have distinct degrees. The locally irregular edge coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of $G$. We say that a multigraph $G$ is locally irregular colorable if it admits a locally irregular edge coloring and we denote by ${\rm lir}(G)$ the locally irregular chromatic index of $G$, which is the smallest number of colors required in a locally irregular edge coloring of a locally irregular colorable multigraph $G$. We conjecture that for every connected graph $G$, which is not isomorphic to $K_2$, multigraph $^2G$ obtained from $G$ by doubling each edge admits ${\rm lir}(^2G)\leq 2$. This concept is closely related to the well known 1-2-3 Conjecture, Local Irregularity Conjecture, (2, 2) Conjecture and other similar problems concerning edge colorings. We show this conjecture holds for graph classes like paths, cycles, wheels, complete graphs, complete $k$-partite graphs and bipartite graphs. We also prove the general bound for locally irregular chromatic index for all 2-multigraphs using our result for bipartite graphs.