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 $F$ be a hereditary family of graphs; i.e., if $H\in F$ and $H^{\prime }\subset H$ then also $H^{\prime }\subset F$. For a graph $G$ and any integer $n\ge \left|G\right|$, let $f(n,G|F)$ 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 $F$.

The case $F=\left\{K_2\right\}$ is the notorious anti-Ramsey rainbow coloring problem introduced by Erdős, Simonovits and Sós in 1973.

Using the $F$-deck of $G$, $D\left(G\right|F)=\{H:H=G-D,D\in F\}$, we define ${\chi }_{{}_F}\left(G\right)=min\{\chi \left(H\right):H\in D\left(G\right|F)\}$.

The main theorem we prove is: Suppose $F$ is a hereditary family of graphs, and let $G$ be a graph not a member of $F$. (1) If ${\chi }_{{}_F}\left(G\right)\ge 3$, then $f(n,G|F)=(1+o\left(1\right))\mathrm{ex}(n,K_{{\chi }_{{}_F}\left(G\right)})$. (2) Otherwise $f(n,G|F)=o\left(n^2\right)$.

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 $p\ge 5$ and $F=\{t{K}_2:t\ge 1\}$, we have $f(n,K_p|F)=(1+o\left(1\right))\mathrm{ex}(n,K_{\lceil p/2\rceil })$; this is the smallest number of colors such that any edge coloring of $K_n$ with this many colors contains 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.