Abstract: Let $K_{n}^{c}$ denote a complete graph on $n$ vertices whose edges are
colored in an arbitrary way. Let $\Delta^{mon}
(K_{n}^{c})$ denote the maximum number of edges of the same color incident with a vertex of $K^c_{n}$. 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 $\Delta^{{mon}}
(K_{n}^{c})<\lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored
Hamiltonian cycle. This conjecture is still open. In this paper, we study properly colored paths and cycles under the condition mentioned in the above conjecture.  

Key words: properly colored cycle, complete graph