Minimal colorings for properly colored subgraphs in complete graphs
Chunqiu Fang, Ervin Gy\H{o}ri, Jimeng Xiao

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 be the maximum number of colors in an edge-coloring of with no properly colored copy of . In this paper, we show that where . Furthermore, we determine the value of for and and the exact value of , where is and , respectively. Also, we give an upper bound and a lower bound of .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph Labeling and Dimension Problems
