Monochromatic graph decompositions inspired by anti-Ramsey theory and the odd-coloring problem

Abstract

We consider extremal edge-coloring problems inspired by the theory of anti-Ramsey / rainbow coloring, and further by odd-colorings and conflict-free colorings. Let $G$ be a graph, and $F$ any given family of graphs. For every integer $n \geq |G|$, let $f(n,G|F)$ denote the smallest integer $k$ such that any edge coloring of the complete graph $K_n$ with at least $k$ colors forces a copy of $G$ in which each color class induces a member of $F$. Observe that in anti-Ramsey problems each color class is a single edge; i.e., $F=\{K_2\}$. In our previous paper [arXiv:2405.19812], attention was given mostly to the case where $F$ is hereditary under subgraph inclusion. In the present work we consider coloring problems inspired by odd-coloring and conflict-free coloring. As we shall see, dealing with these problems requires distinct additional tools to those used in our first paper on the subject. Among the many results introduced in this paper, we mention: (1) For every graph $G$, there exists a constant $c=c(G)$ such that in any edge coloring of $K_n$ with at least $cn$ colors there is a copy of $G$ in which every vertex $v$ is incident with an edge whose color appears only once among all edges incident with $v$. (2) In sharp contrast to the above result we prove that if $F$ is the class of all odd graphs (having vertices with odd degrees only) then $f(n,K_k|F)=(1+o(1))$ex$(n,K_{\lceil k/2 \rceil})$, which is quadratic for $k \geq 5$. (3) We exactly determine $f(n,G|F)$ for small graphs when $F$ belongs to several families representing various odd/even coloring constraints.