Decomposing graphs into a constant number of locally irregular subgraphs

TL;DR

This paper establishes a constant upper bound on the irregular chromatic index for bipartite graphs, advancing the understanding of graph decompositions into locally irregular subgraphs and supporting a broader conjecture.

Contribution

It proves that bipartite graphs (except odd-length paths) have an irregular chromatic index at most 10, the first such bound for bipartite graphs, and extends results to general graphs and highly connected bipartite graphs.

Findings

Bipartite graphs (not odd paths) have an irregular chromatic index ≤ 10.

Every graph with a locally irregular decomposition has an index ≤ 328.

16-edge-connected bipartite graphs have an index ≤ 2.

Abstract

A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index $\chi_{\rm irr}'(G)$ of a graph $G$ is the smallest number of locally irregular subgraphs needed to edge-decompose $G$. Not all graphs have such a decomposition, but Baudon, Bensmail, Przyby{\l}o, and Wo\'zniak conjectured that if $G$ can be decomposed into locally irregular subgraphs, then $\chi_{\rm irr}'(G)\leq 3$. In support of this conjecture, Przyby{\l}o showed that $\chi_{\rm irr}'(G)\leq 3$ holds whenever $G$ has minimum degree at least $10^{10}$. Here we prove that every bipartite graph $G$ which is not an odd length path satisfies $\chi_{\rm irr}'(G)\leq 10$. This is the first general constant upper bound on the irregular chromatic index of bipartite graphs. Combining this result with Przyby{\l}o's result, we show that $\chi_{\rm irr}'(G) \leq 328$ for every graph $G$ which admits a decomposition into locally irregular subgraphs. Finally, we show that $\chi_{\rm irr}'(G)\leq 2$ for every $16$-edge-connected bipartite graph $G$.