Long induced paths and forbidden patterns: Polylogarithmic bounds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912836039475200 |
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| author | Duron, Julien Esperet, Louis Raymond, Jean-Florent |
| author_facet | Duron, Julien Esperet, Louis Raymond, Jean-Florent |
| contents | Consider a graph $G$ with a long path $P$. When is it the case that $G$ also contains a long induced path? This question has been investigated in general as well as within a number of different graph classes since the 80s. We have recently observed in a companion paper (Long induced paths in sparse graphs and graphs with forbidden patterns, arXiv:2411.08685, 2024) that most existing results can recovered in a simple way by considering forbidden ordered patterns of edges along the path $P$. In particular we proved that if we forbid some fixed ordered matching along a path of order $n$ in a graph $G$, then $G$ must contain an induced path of order $(\log n)^{Ω(1)}$. Moreover, we completely characterized the forbidden ordered patterns forcing the existence of an induced path of polynomial size.
The purpose of the present paper is to completely characterize the ordered patterns $H$ such that forbidding $H$ along a path $P$ of order $n$ implies the existence of an induced path of order $(\log n)^{Ω(1)}$. These patterns are star forests with some specific ordering, which we called constellations.
As a direct consequence of our result, we show that if a graph $G$ has a path of length $n$ and does not contain $K_t$ as a topological minor, then $G$ contains an induced path of order $(\log n)^{Ω(1/t \log^2 t)}$. The previously best known bound was $(\log n)^{f(t)}$ for some unspecified function $f$ depending on the Topological Minor Structure Theorem of Grohe and Marx (2015). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long induced paths and forbidden patterns: Polylogarithmic bounds Duron, Julien Esperet, Louis Raymond, Jean-Florent Discrete Mathematics Combinatorics Consider a graph $G$ with a long path $P$. When is it the case that $G$ also contains a long induced path? This question has been investigated in general as well as within a number of different graph classes since the 80s. We have recently observed in a companion paper (Long induced paths in sparse graphs and graphs with forbidden patterns, arXiv:2411.08685, 2024) that most existing results can recovered in a simple way by considering forbidden ordered patterns of edges along the path $P$. In particular we proved that if we forbid some fixed ordered matching along a path of order $n$ in a graph $G$, then $G$ must contain an induced path of order $(\log n)^{Ω(1)}$. Moreover, we completely characterized the forbidden ordered patterns forcing the existence of an induced path of polynomial size. The purpose of the present paper is to completely characterize the ordered patterns $H$ such that forbidding $H$ along a path $P$ of order $n$ implies the existence of an induced path of order $(\log n)^{Ω(1)}$. These patterns are star forests with some specific ordering, which we called constellations. As a direct consequence of our result, we show that if a graph $G$ has a path of length $n$ and does not contain $K_t$ as a topological minor, then $G$ contains an induced path of order $(\log n)^{Ω(1/t \log^2 t)}$. The previously best known bound was $(\log n)^{f(t)}$ for some unspecified function $f$ depending on the Topological Minor Structure Theorem of Grohe and Marx (2015). |
| title | Long induced paths and forbidden patterns: Polylogarithmic bounds |
| topic | Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2412.14863 |