Long monochromatic paths and cycles in 2-edge-colored multipartite graphs
| Autor: | Balogh, József, Kostochka, Alexandr, Lavrov, Mikhail, Liu, Xujun |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We solve four similar problems: For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic (i) cycle $C_{2n}$ with $2n$ vertices, (ii) cycle $C_{\geq 2n}$ with at least $2n$ vertices, (iii) path $P_{2n}$ with $2n$ vertices, and (iv) path $P_{2n+1}$ with $2n+1$ vertices. This implies a generalization for large $n$ of the conjecture by Gy\'arf\'as, Ruszink\'o, S\'ark\H{o}zy and Szemer\'edi that for every $2$-edge-coloring of the complete $3$-partite graph $K_{n,n,n}$ there is a monochromatic path $P_{2n+1}$. An important tool is our recent stability theorem on monochromatic connected matchings. Comment: 46 pages, 4 figures |
| Databáze: | arXiv |
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