Decomposability of regular graphs to 4 locally irregular subgraphs

Jakub Przybyło

Applied Mathematics and Computation, 2024, vol. 480, issue C

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≥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.

Keywords: Locally irregular graph; Graph decomposition; Edge set partition (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:480:y:2024:i:c:s0096300324003771

DOI: 10.1016/j.amc.2024.128916

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