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arXiv · 2402.18739

Decomposability of regular graphs to $4$ locally irregular subgraphs

Jakub Przybyło
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Abstract

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. It was conjectured that every connected graph is edge decomposable to $3$ locally irregular subgraphs, unless it belongs to a certain family of exceptions, including graphs of small maximum degrees, which are not decomposable to any number of such subgraphs. Recently Sedlar and Škrekovski exhibited a counterexample to the conjecture, which necessitates a decomposition to (at least) $4$ locally irregular subgraphs. We prove that every $d$-regular graph with $d$ large enough, i.e. $d\geq 54000$, is decomposable to $4$ locally irregular subgraphs. Our proof relies on a mixture of a numerically optimized application of the probabilistic method and certain deterministic results on degree constrained subgraphs due to Addario-Berry, Dalal, McDiarmid, Reed, and Thomason, and to Alon and Wei, introduced in the context of related problems concerning irregular subgraphs.

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Jakub Przybyło. 2024-02-28. Decomposability of regular graphs to $4$ locally irregular subgraphs. https://arxiv.org/abs/2402.18739

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