Finding Large Monochromatic Diameter Two Subgraphs

TL;DR

This paper investigates the size of the largest monochromatic diameter two subgraphs in edge-colored complete graphs, establishing optimal bounds for different numbers of colors.

Contribution

It proves tight bounds for the maximum size of monochromatic diameter two subgraphs in k-colored complete graphs, including exact results for two and three or more colors.

Findings

For k ≥ 3, the bound is asymptotically optimal.

For k=2, a monochromatic diameter two subgraph of at least 3/4 of the vertices always exists.

These bounds are proven to be tight and optimal.

Abstract

Given a coloring of the edges of the complete graph on n vertices in k colors, by considering the neighbors of an arbitrary vertex it follows that there is a monochromatic diameter two subgraph on at least 1+(n-1)/k vertices. We show that for $k \ge 3$ this is asymptotically best possible, and that for k=2 there is always a monochromatic diameter two subgraph on at least $\lceil {3 \over 4} n \rceil$ vertices, which again, is best possible.