Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings
Yair Caro, Zsolt Tuza
TL;DR
This work completes a trilogy that generalizes anti-Ramsey theory by introducing the $f$-function and the piercing $g$-function for a graph $G$ with respect to a graph family $\mathcal{F}$. It develops a robust framework combining extremal and packing methods, canonical colorings, and structural exclusions to derive asymptotically tight results across a broad spectrum of $\mathcal{F}$ and $G$, including quadratic and subquadratic growth regimes. Key contributions include a detailed treatment of how exclusions shape growth, stability results for chromatic parameters, and the piercing problem via graph packing, blockers, and anti-packers, with extensive exact and asymptotic results for $K_p$ and various $\mathcal{F}$. The findings illuminate strong connections between original anti-Ramsey numbers, Turán-type extremal theory, and recent packing techniques, offering a versatile toolkit for future investigations into generalized rainbow problems and their applications to graphs and beyond.
Abstract
Anti-Ramsey theory was initiated in 1975 by Erdős, Simonovits and Sós, inspiring hundreds of publications since then. The present work is the third and last piece of our trilogy in which we introduce a far-reaching generalization via the following two functions for any graph $G$ and family ${\cal F}$ of graphs: If $K_2 \in {\cal F}$, let $f(n,G|{\cal F})$ be the smallest integer $k$ such that every edge coloring of $K_n$ with at least $k$ colors forces a copy of $G$ in which all color classes are members of ${\cal F}$. If $K_2 \notin {\cal F}$, let $g(n,G|{\cal F})$ be the largest integer $k$ for which there exists an edge coloring of $K_n$ using exactly $k$ colors, such that every copy of $G$ contains an induced color class which is a member of ${\cal F}$. We develop methods suitable for deriving asymptotically tight results for the $f$-function and the $g$-function for many combinations of $G$ and ${\cal F}$. The preceding parts of the trilogy are arXiv: 2405.19812 and 2408.04257, published in Discrete Applied Math. Vol. 363 and Mathematics Vol. 12:23, respectively.