On better-quasi-ordering under graph minors
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912663966056448 |
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| author | Georgakopoulos, Agelos |
| author_facet | Georgakopoulos, Agelos |
| contents | In the aftermath of the Robertson--Seymour Graph Minor Theorem, Thomas conjectured that the countable graphs are well-quasi-ordered under the minor relation. We prove that this conjecture, when restricted to graphs with no infinite paths (rays), is equivalent to the statement that the finite graphs are better-quasi-ordered, another well-known open problem. Even more, we prove that the latter implies that the countable rayless graphs are better-quasi-ordered.
We prove several other statements to be equivalent to the above, one of which being that the rayless countable graphs of rank $α$ can be decomposed into exactly $\aleph_0$ minor-twin classes for every ordinal $α<ω_1$.
By restricting the latter statement to trees, and combining it with Nash-Williams' theorem that the infinite trees are well-quasi-ordered, we deduce as a side result that a minor-closed family of N-labelled rayless forests is Borel -- in the Tychonoff product topology -- if and only if it does not contain all rayless forests.
As another side-result, we prove Seymour's self-minor conjecture for rayless graphs of any cardinality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On better-quasi-ordering under graph minors Georgakopoulos, Agelos Combinatorics Logic 05C83, 05C63, 06A07 In the aftermath of the Robertson--Seymour Graph Minor Theorem, Thomas conjectured that the countable graphs are well-quasi-ordered under the minor relation. We prove that this conjecture, when restricted to graphs with no infinite paths (rays), is equivalent to the statement that the finite graphs are better-quasi-ordered, another well-known open problem. Even more, we prove that the latter implies that the countable rayless graphs are better-quasi-ordered. We prove several other statements to be equivalent to the above, one of which being that the rayless countable graphs of rank $α$ can be decomposed into exactly $\aleph_0$ minor-twin classes for every ordinal $α<ω_1$. By restricting the latter statement to trees, and combining it with Nash-Williams' theorem that the infinite trees are well-quasi-ordered, we deduce as a side result that a minor-closed family of N-labelled rayless forests is Borel -- in the Tychonoff product topology -- if and only if it does not contain all rayless forests. As another side-result, we prove Seymour's self-minor conjecture for rayless graphs of any cardinality. |
| title | On better-quasi-ordering under graph minors |
| topic | Combinatorics Logic 05C83, 05C63, 06A07 |
| url | https://arxiv.org/abs/2510.19285 |