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 $np=he^h$. We show that, with high probability, every $2$-edge-coloring of $G$ contains a monochromatic connected component of order at least $n-\Theta(ne^{-h})$. 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 $\frac{n}{2}-\Theta(1/p)$, 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 $\Theta(ne^{-h})$ vertices, and this bound is asymptotically sharp. Our approach is elementary and is based on analyzing large connected structures across suitably balanced vertex partitions.
A path in a properly edge-colored graph is rainbow if its edges have pairwise distinct colors. For a proper edge-coloring $c$ of a graph $G$, let $\operatorname{rpc}(G,c)$ be the minimum number of rainbow paths needed to cover $E(G)$, and let $\operatorname{rpc}(G)$ be the maximum of $\operatorname{rpc}(G,c)$ over all...
We prove that there is an absolute constant $c>0$ such that every graph of chromatic number at least $r$ and at most $cr^3\log^2 r$ edges contains at least $\binom r3$ triangles. The proof has three ingredients. First, a sparse-core argument based on a triangle-sensitive coloring estimate of Harris extracts, from any c...
We show that almost every graph admits a partition of its vertex set into three parts such that no two adjacent vertices have the same number of neighbors in each of the three parts. Equivalently, for $G\sim G(n,1/2)$, $\chi_m(G)\le3$ with high probability, improving the previously known bound of five. Here $\chi_m(G)$...
Let $H \sim H^{k}_c(n,p)$ be an edge-colored random $k$-uniform hypergraph on the vertex set $[n]$, where each edge $e \in \binom{[n]}{k}$ is included independently with probability $p$ and is uniformly and independently assigned a color from the color set $[c]$. For $k = 2$, Ferber and Krivelevich (2016) established t...
For a graph $G$ of order $n$, put $\sigma(G)=(\lambda_1(G)+\lambda_n(G))/n$. We determine the first three largest values of this invariant among nonbipartite distance-regular graphs of diameter three and odd girth at least seven. The unique maximizer is the folded $7$-cube, with value $1/32$; the unique second maximize...
An equitable $k$-coloring of a graph partitions its vertex set into $k$ independent sets whose sizes differ by at most one; the least such $k$ is the equitable chromatic number $\chie(G)$. Every known bound on $\chie$ valid for all graphs, beginning with the Hajnal--Szemer\'edi theorem, is linear in the maximum degree...
Mohammad F. Marashdeh· 0 citations
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