Long monochromatic paths and cycles in 2-colored bipartite graphs

Abstract

Gyárfás and Lehel and independently Faudree and Schelp proved that in any 2-coloring of the edges of Kn,n there exists a monochromatic path on at least 2⌈n∕2⌉ 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, for every γ≫η>0 and n≥n0(γ), if G is a balanced bipartite graph on 2n vertices with minimum degree at least (3∕4+γ)n, then in every 2-coloring of the edges of G, either there exists a monochromatic cycle on at least (1+η)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⌈n∕2⌉ vertices and a monochromatic cycle on at least 2⌊n∕2⌋ vertices. Furthermore, we determine an asymptotically tight bound on the length of a longest monochromatic cycle in a 2-colored balanced bipartite graph on 2n vertices with minimum degree δn for all 0≤δ≤1.