On locally irregular decompositions of subcubic graphs

Abstract

A graph $G$ is locally irregular if every two adjacent vertices of $G$ have different degrees. A locally irregular decomposition of $G$ is a partition $E_1,\dots,E_k$ of $E(G)$ such that each $G[E_{i}]$ is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, in that sense, it was conjectured by Baudon, Bensmail, Przybyło and Woźniak that they decompose into at most 3 locally irregular graphs. Towards that conjecture, it was recently proved by Bensmail, Merker and Thomassen that every decomposable graph decomposes into at most 328 locally irregular graphs. We here focus on locally irregular decompositions of subcubic graphs, which form an important family of graphs in this context, as all non-decomposable graphs are subcubic. As a main result, we prove that decomposable subcubic graphs decompose into at most 5 locally irregular graphs, and only at most 4 when the maximum average degree is less than $\frac{12}{5}$. We then consider weaker decompositions, where subgraphs can also include regular connected components, and prove the relaxations of the conjecture above for subcubic graphs.