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Decomposability of regular graphs to 4 locally irregular subgraphs

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  • Przybyło, Jakub

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.

Suggested Citation

  • Przybyło, Jakub, 2024. "Decomposability of regular graphs to 4 locally irregular subgraphs," Applied Mathematics and Computation, Elsevier, vol. 480(C).
    • Handle: RePEc:eee:apmaco:v:480:y:2024:i:c:s0096300324003771
    DOI: 10.1016/j.amc.2024.128916
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    References listed on IDEAS

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    1. Hui Lei & Xiaopan Lian & Yongtang Shi & Ran Zhao, 2022. "Graph Classes with Locally Irregular Chromatic Index at most 4," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 903-918, December.
    2. Jelena Sedlar & Riste Škrekovski, 2021. "Remarks on the Local Irregularity Conjecture," Mathematics, MDPI, vol. 9(24), pages 1-10, December.
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