Monochromatic graph decompositions inspired by anti-Ramsey colorings

Abstract

We consider coloring problems inspired by the theory of anti-Ramsey / rainbow colorings that we generalize to a far extent. Let $\mathcal{F}$ be a hereditary family of graphs; i.e., if $H\in \mathcal{F}$ and $H'\subset H$ then also $H'\subset \mathcal{F}$. For a graph $G$ and any integer $n \geq |G|$, let $f(n,G|\mathcal{F})$ denote the smallest number $k$ of colors such that any edge coloring of $K_n$ with at least $k$ colors forces a copy of $G$ in which each color class induces a member of $\mathcal{F}$. The case $\mathcal{F} = \{K_2\}$ is the notorious anti-Ramsey / rainbow coloring problem introduced by Erdős, Simonovits and Sós in 1973. Using the $\mathcal{F}$-deck of $G$, $D(G|\mathcal{F}) = \{ H : H = G - D, \, D \in \mathcal{F}\}$, we define $χ_\mathcal{F}(G) = \min \{ χ(H) : H \in D(G|\mathcal{F}) \}$. The main theorem we prove is: Suppose $\mathcal{F}$ is a hereditary family of graphs, and let $G$ be a graph not a member of $\mathcal{F}$. (1) If $χ_\mathcal{F}(G) \geq 3$, then $f(n, G |\mathcal{F}) = (1+o(1)) \, ex(n, K_{χ_\mathcal{F}(G)})$. (2) Otherwise $f(n, G |\mathcal{F}) = o(n^2)$. Among the families covered by this theorem are: matchings, acyclic graphs, planar and outerplanar graphs, $d$-degenerate graphs, graphs with chromatic number at most $k$, graphs with bounded maximum degree, and many more. We supply many concrete examples to demonstrate the wide range of applications of the main theorem; the next result is a representative of these examples. For $p \geq 5$ and $\mathcal{F} = \{ tK_2 : t \geq 1 \}$, we have $f(n,K_p |\mathcal{F}) = (1+o(1)) \, ex(n, K_{\lceil p/2 \rceil})$; this means a properly colored copy of $K_p$. In other words, a certain number of colors forces nearly twice as large properly edge-colored complete subgraphs as rainbow ones.