On the twin-width of near-regular graphs
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914031013462016 |
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| author | Heinrich, Irene Ihringer, Ferdinand Raßmann, Simon Volk, Lena |
| author_facet | Heinrich, Irene Ihringer, Ferdinand Raßmann, Simon Volk, Lena |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_02342 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the twin-width of near-regular graphs Heinrich, Irene Ihringer, Ferdinand Raßmann, Simon Volk, Lena Combinatorics Data Structures and Algorithms 68Q25, 68R10, 68R05 F.2.2; G.2.1; G.2.2 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. |
| title | On the twin-width of near-regular graphs |
| topic | Combinatorics Data Structures and Algorithms 68Q25, 68R10, 68R05 F.2.2; G.2.1; G.2.2 |
| url | https://arxiv.org/abs/2504.02342 |