Monochromatic graph decompositions inspired by anti-Ramsey colorings

TL;DR

This paper generalizes anti-Ramsey coloring problems to hereditary graph families, establishing thresholds for the number of colors needed to force specific subgraphs with certain properties, and demonstrates wide applicability across various graph classes.

Contribution

It introduces a broad framework for anti-Ramsey problems involving hereditary graph families and provides a main theorem linking color thresholds to Turán numbers, with numerous applications.

Findings

For certain graph families, the minimum number of colors needed is asymptotically the Turán number.

Properly colored complete subgraphs can be forced with fewer colors than rainbow ones.

The results apply to diverse graph classes like matchings, planar graphs, and bounded degree graphs.

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 Erd\H{o}s, Simonovits and S\'os in 1973. Using the $\mathcal{F}$-deck of $G$, $D(G|\mathcal{F}) = \{ H : H = G - D, \, D \in \mathcal{F}\}$, we define $\chi_\mathcal{F}(G) = \min \{ \chi(H) : H \in D(G|\mathcal{F}) \}$. 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 $\chi_\mathcal{F}(G) \geq 3$, then $f(n, G |\mathcal{F}) = (1+o(1)) \, ex(n, K_{\chi_\mathcal{F}(G)})$. (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.