Highly connected monochromatic subgraphs

Abstract

AbstractWe conjecture that for n>4(k-1) every 2-coloring of the edges of the complete graph Kn contains a k-connected monochromatic subgraph with at least n-2(k-1) vertices. This conjecture, if true, is best possible. Here we prove it for k=2, and show how to reduce it to the case n<7k-6. We prove the following result as well: for n>16k every 2-colored Kn contains a k-connected monochromatic subgraph with at least n-12k vertices