Forbidden Properly Edge-Colored Subgraphs that Force Large Highly Connected Monochromatic Subgraphs

Abstract

We consider the connected graphs G that satisfy the following property: If $n \gg m \gg k$ are integers, then any coloring of the edges of $K_{n}$, using m colors, containing no properly colored copy of G, contains a monochromatic k-connected subgraph of order at least $n - f(G, k, m)$ where f does not depend on n. If we let $\mathscr {G}$ denote the set of graphs satisfying this statement, we exhibit some infinite families of graphs in $\mathscr {G}$ as well as conjecture that the cycles in $\mathscr {G}$ are precisely those whose lengths are divisible by 3. Our main result is that $C_{6} \in \mathscr {G}$.