Let Kncβ denote a complete graph on n vertices whose edges are
colored in an arbitrary way. Let Ξmon(Kncβ) denote the
maximum number of edges of the same color incident with a vertex of
Kncβ. A properly colored cycle (path) in Kncβ 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 Ξmon(Kncβ)<β2nββ, then Kncβ contains a properly
colored Hamiltonian cycle. Li, Wang and Zhou proved that if
Ξmon(Kncβ)<β2nββ, then
Kncβ contains a properly colored cycle of length at least β3n+2ββ+1. In this paper, we improve the bound to β2nββ+2.Comment: 8 page