Properly colored cycles in edge-colored complete graphs

Abstract

As an analogy of the well-known anti-Ramsey problem, we study the existence of properly colored cycles of given length in an edge-colored complete graph. Let $\mathrm{pr}(K_n,G)$ be the maximum number of colors in an edge-coloring of $K_n$ with no properly colored copy of G. In this paper, we determine the exact threshold for cycles $\mathrm{pr}(K_n,C_{\ell })$, which proves a conjecture proposed by Fang, Győri, and Xiao, that the maximum number of colors in an edge-coloring of $K_n$ with no properly colored copy of $C_{\ell }$ is $\max \{\left(\begin{array}{c}\ell -1\\ 2\end{array}\right)+n-\ell +1,\lfloor \frac{\ell -1}{3}\rfloor n-\left(\begin{array}{c}\lfloor \frac{\ell -1}{3}\rfloor +1\\ 2\end{array}\right)+1+r_{\ell -1}\}$, where $C_{\ell }$ is a cycle on ℓ vertices, $\ell -1\equiv r_{\ell -1}\mathrm{mod}3$, and $0\le r_{\ell -1}\le 2$. It is a slight modification of a previous conjecture posed by Manoussakis, Spyratos, Tuza and Voigt. Also, we consider the maximal coloring of $K_n$ whether a properly colored cycle can be extended by exact one more vertex.