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arXiv · 2504.02342

On the twin-width of near-regular graphs

Irene Heinrich·Ferdinand Ihringer·Simon Raßmann·Lena Volk
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Abstract

Twin-width is a recently introduced graph parameter based on the repeated contraction of near-twins. It has shown remarkable utility in algorithmic and structural graph theory, as well as in finite model theory -- particularly since first-order model checking is fixed-parameter tractable when a witness certifying small twin-width is provided. However, the behavior of twin-width in specific graph classes, particularly cubic graphs, remains poorly understood. While cubic graphs are known to have unbounded twin-width, no explicit cubic graph of twin-width greater than 4 is known. This paper explores this phenomenon in regular and near-regular graph classes. We show that extremal graphs of bounded degree and high twin-width are asymmetric, partly explaining their elusiveness. Additionally, we establish bounds for circulant and d-degenerate graphs, and examine strongly regular graphs, which exhibit similar behavior to cubic graphs. Our results include determining the twin-width of Johnson graphs over 2-sets, and cyclic Latin square graphs.

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Irene Heinrich, Ferdinand Ihringer, Simon Raßmann, Lena Volk. 2025-04-03. On the twin-width of near-regular graphs. https://doi.org/10.1016/j.dam.2025.07.044

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