Large Monochromatic Components in Colored Random Graphs

Abstract

We study the size of the largest monochromatic connected component that must appear in any edge-coloring of a random graph. Let $G\sim G(n,p)$ with $p\gg 1/n$ and $p=o(1)$, and write $np=h{e}^h$. We show that, with high probability, every $2$-edge-coloring of $G$ contains a monochromatic connected component of order at least $n-\hat{I}˜(n{e}^{-h})$. Moreover, we construct colorings showing that this bound is best possible up to constant factors. We extend this result to three colors: for $p\gg 1/n$ and $p=o(1)$, with high probability every $3$-edge-coloring of $G$ contains a monochromatic connected component of size at least $\frac{n}{2}-\hat{I}˜(1/p)$, and this estimate is again tight up to constant factors. In the bipartite setting $G\sim G(n,n,p)$, under the same assumptions on $p$, we prove an analogous statement: with high probability, every $2$-edge-coloring contains two monochromatic components whose union covers all but $\hat{I}˜(n{e}^{-h})$ vertices, and this bound is asymptotically sharp. Our approach is elementary and is based on analyzing large connected structures across suitably balanced vertex partitions.

16 pages