Long monochromatic paths and cycles in 2colored bipartite graphs
Abstract
Gyárfás and Lehel and independently Faudree and Schelp proved that in any 2coloring of the edges of $K_{n,n}$ there exists a monochromatic path on at least $2\lceil n/2\rceil$ vertices, and this is tight. We prove a stability version of this result which holds even if the host graph is not complete; that is, if $G$ is a balanced bipartite graph on $2n$ vertices with minimum degree at least $(3/4+o(1))n$, then in every 2coloring of the edges of $G$, either there exists a monochromatic cycle on at least $(1+o(1))n$ vertices, or the coloring of $G$ is close to an extremal coloring  in which case $G$ has a monochromatic path on at least $2\lceil n/2\rceil$ vertices and a monochromatic cycle on at least $2\lfloor n/2\rfloor$ vertices. Furthermore, we determine an asymptotically tight bound on the length of a longest monochromatic cycle in a 2colored balanced bipartite graph on $2n$ vertices with minimum degree $\delta n$ for all $0\leq \delta\leq 1$.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 DOI:
 10.48550/arXiv.1806.05119
 arXiv:
 arXiv:1806.05119
 Bibcode:
 2018arXiv180605119D
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 18 pages, 2 figures