2-Factors in Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914538330259456 |
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| author | Heuvel, Jan van den Toft, Bjarne |
| author_facet | Heuvel, Jan van den Toft, Bjarne |
| contents | An account of 2-factors in graphs and their history is presented. We give a direct graph-theoretic proof of the 2-Factor Theorem and a new variant of it, and also a new complete characterisation of the maximal graphs without 2-factors. This is based on the important works of Tibor Gallai on 1-factors and of Hans-Boris Belck on k-factors, both published in 1950 and independently containing the theory of alternating chains. We also present an easy proof that a $(2k+1)$-regular graph with at most $2k$ leaves has a 2-factor, and we describe all connected $(2k+1)$-regular graphs with exactly $2k+1$ leaves without a 2-factor. This generalises Julius Petersen's famous theorem, that any 3-regular graph with at most two leaves has a 1-factor, and it generalises the extremal graphs Sylvester discovered for that theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_11486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | 2-Factors in Graphs Heuvel, Jan van den Toft, Bjarne Combinatorics 05C70, 05-03 An account of 2-factors in graphs and their history is presented. We give a direct graph-theoretic proof of the 2-Factor Theorem and a new variant of it, and also a new complete characterisation of the maximal graphs without 2-factors. This is based on the important works of Tibor Gallai on 1-factors and of Hans-Boris Belck on k-factors, both published in 1950 and independently containing the theory of alternating chains. We also present an easy proof that a $(2k+1)$-regular graph with at most $2k$ leaves has a 2-factor, and we describe all connected $(2k+1)$-regular graphs with exactly $2k+1$ leaves without a 2-factor. This generalises Julius Petersen's famous theorem, that any 3-regular graph with at most two leaves has a 1-factor, and it generalises the extremal graphs Sylvester discovered for that theorem. |
| title | 2-Factors in Graphs |
| topic | Combinatorics 05C70, 05-03 |
| url | https://arxiv.org/abs/2510.11486 |