Long properly colored cycles in edge colored complete graphs

Let Kc n denote a complete graph on n vertices whose edges are colored in an arbitrary way. Let ∆mon(Kc n) denote the maximum number of edges of the same color incident with a vertex of Kn. A properly colored cycle (path) in Kc n is a cycle (path) in which adjacent edges have distinct colors. B. Bollobás and P. Erdös (1976) proposed the following conjecture: If ∆mon(Kc n) < b 2 c, then Kc n contains a properly colored Hamiltonian cycle. Li, Wang and Zhou proved that if ∆mon(Kc n) < b 2 c, then Kc n contains a properly colored cycle of length at least d n+2 3 e + 1. In this paper, we improve the bound to d 2 e + 2.