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Long monochromatic paths and cycles in 2-edge-colored multipartite graphs
- Publication Year :
- 2019
-
Abstract
- 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.<br />Comment: 46 pages, 4 figures
- Subjects :
- Mathematics - Combinatorics
05C15, 05C35, 05C38
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1905.04657
- Document Type :
- Working Paper