Long induced paths in sparse graphs and graphs with forbidden patterns
| Autor: | Duron, Julien, Esperet, Louis, Raymond, Jean-Florent |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | Consider a graph $G$ with a path $P$ of order $n$. What conditions force $G$ to also have a long induced path? As complete bipartite graphs have long paths but no long induced paths, a natural restriction is to forbid some fixed complete bipartite graph $K_{t,t}$ as a subgraph. In this case we show that $G$ has an induced path of order $(\log \log n)^{1/5-o(1)}$. This is an exponential improvement over a result of Galvin, Rival, and Sands (1982) and comes close to a recent upper bound of order $O((\log \log n)^2)$. Another way to approach this problem is by viewing $G$ as an ordered graph (where the vertices are ordered according to their position on the path $P$). From this point of view it is most natural to consider which ordered subgraphs need to be forbidden in order to force the existence of a long induced path. Focusing on the exclusion of ordered matchings, we improve or recover a number of existing results with much simpler proofs, in a unified way. We also show that if some forbidden ordered subgraph forces the existence of a long induced path in $G$, then this induced path has size at least $\Omega((\log \log \log n)^{1/3})$, and can be chosen to be increasing with respect to $P$. Comment: 26 pages, 8 figures. Comments welcome ! |
| Databáze: | arXiv |
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