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Lecture on Monochromatic Connected Matchings, Paths and Cycles in 2-Edge-Colored Multipartite Graphs
DateandTime: 2019-08-11 10:22:31

Speaker:Xujun Liu,University of Illinois Urbana Champaign (UIUC)

Sponsor:Professor Wang Guanghui

Date:August. 13, 2019

Time:9:00 am

Location:Conference Hall 1032, Zone B, Zhixin Building, Central Campus


We solve four similar problems: For every fixed s and large n, we describe all values of n1, …, ns such that for every 2-edge-coloring of the complete s-partite graph Kn1,...,ns there exists a monochromatic (i) cycle C2n with 2n vertices, (ii) cycle C≥2n with at least 2n vertices, (iii) path P2n with 2n vertices, and (iv) path P2n+1 with 2n + 1 vertices. This implies a generalization of the conjecture by Gy´arf´as, Ruszink´o, S´ark˝ozy and Szemer´edi that for every 2-edge-coloring of the complete 3-partite graph Kn,n,n there is a monochromatic path P2n+1. An important tool is our recent stability theorem on monochromatic connected matchings (A matching M in G is connected if all the edges of M are in the same component of G). We will also talk about exact Ramsey-type bounds on the sizes of monochromatic connected matchings in 2-colored multipartite graphs. Moreover, we will talk about the existence of long monochromatic paths and cycles in 2-edge-colored graphs with large minimum degree. Joint work with J´ozsef Balogh, Alexandr Kostochka and Mikhail Lavrov.

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Edited by Zhang Hao

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