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 Δ mon ( 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. Bollobás and P. Erdös (1976) proposed the following conjecture: if Δ mon ( K n c ) < ⌊ n 2 ⌋ , then K n c contains a properly colored Hamiltonian cycle. Li, Wang and Zhou proved that if Δ mon ( K n c ) < ⌊ n 2 ⌋ , then K n c contains a properly colored cycle of length at least ⌈ n + 2 3 ⌉ + 1 . In this paper, we improve the bound to ⌈ n 2 ⌉ + 2 .