Minimal colorings for properly colored subgraphs in complete graphs

TL;DR

This paper investigates the maximum number of colors in edge-colorings of complete graphs avoiding properly colored subgraphs, providing asymptotic relations, exact values for certain graphs, and bounds for others.

Contribution

It establishes a relation between $pr(K_{n}, G)$ and extremal functions, determines exact values for specific graphs, and offers bounds for complex cases.

Findings

Asymptotic equivalence of $pr(K_{n}, G)$ and extremal functions.

Exact values of $pr(K_{n}, P_{l})$ for $l \\ge 27$ and $n \\ge 2l^{3}$.

Exact values for $G$ being $C_{5}$, $C_{6}$, and $K_{4}^{-}$.

Abstract

Let $pr(K_{n}, G)$ be the maximum number of colors in an edge-coloring of $K_{n}$ with no properly colored copy of $G$. In this paper, we show that $pr(K_{n}, G)-ex(n, \mathcal{G'})=o(n^{2}), $ where $\mathcal{G'}=\{G-M: M \text{ is a matching of }G\}$. Furthermore, we determine the value of $pr(K_{n}, P_{l})$ for $l\ge 27$ and $n\ge 2l^{3}$ and the exact value of $pr(K_{n}, G)$, where $G$ is $C_{5}, C_{6}$ and $K_{4}^{-}$, respectively. Also, we give an upper bound and a lower bound of $pr(K_{n}, K_{2,3})$.