On connected components with many edges

TL;DR

This paper establishes bounds on the size of monochromatic connected subgraphs in edge-colored complete graphs, confirming a conjecture and extending results to multiple colors.

Contribution

It proves a new inequality relating edges in subgraphs of complete multipartite graphs and applies it to colorings, confirming a conjecture on monochromatic connected subgraphs.

Findings

Every 3-coloring contains a monochromatic connected subgraph with at least 1/6 of the edges.

Such colorings have a monochromatic circuit with about 1/6 of the edges.

For k colors, a monochromatic connected subgraph with at least 1/(k^2 - k + 5/4) of the edges exists.

Abstract

We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then $H$ contains a connected component $H'$ satisfying $|E(H')||E(G)|\geq |E(H)|^2$. We use this to prove that every three-coloring of the edges of a complete graph contains a monochromatic connected subgraph with at least $1/6$ of the edges. We further show that such a coloring has a monochromatic circuit with a fraction $1/6-o(1)$ of the edges. This verifies a conjecture of Conlon and Tyomkyn. Moreover, for general $k$, we show that every $k$-coloring of the edges of $K_n$ contains a monochromatic connected subgraph with at least $\frac{1}{k^2-k+\frac{5}{4}}\binom{n}{2}$ edges.