Long properly colored cycles in edge colored complete graphs

Let $K_n^c$ denote a complete graph on $n$ vertices whose edges are colored in an arbitrary way. Let ${\mathrm{\Delta }}^{\mathrm{m}\mathrm{o}\mathrm{n}}(K_n^c)$ denote the maximum number of edges of the same color incident with a vertex of $K_n^c$. A properly colored cycle (path) in $K_n^c$ is a cycle (path) in which adjacent edges have distinct colors. B. Bollob'{a}s and P. Erd"{o}s (1976) proposed the following conjecture: if $\Delta^{$\mathrm{m}\mathrm{o}\mathrm{n}$} (K_{n}^{c})<\lfloor \frac{n}{2} \rfloor$,then$K_{n}^{c}$ contains a properly colored Hamiltonian cycle. Li, Wang and Zhou proved that if ${\mathrm{\Delta }}^{\mathrm{m}\mathrm{o}\mathrm{n}}(K_n^c)<\lfloor \frac{n}{2}\rfloor$, then $K_n^c$ contains a properly colored cycle of length at least $\lceil \frac{n+2}{3}\rceil+1$\mathrm{.}Inthispaper,weimprovetheboundto$\lceil \frac{n}{2}\rceil + 2$.