完全图中的正常染色的路和圈

令$K_{n}^{c}$表示$n$ 个顶点的边染色完全图.
令 $Delta^{mon}
(K_{n}^{c})$表示$K^c_{n}$的顶点上关联的同种颜色的边的最大数目.
如果$K_{n}^{c}$中的一个圈（路）上相邻的边染不同颜色,则称它为正常染色的.
B. Bollob'{a}s和P. Erd"{o}s (1976) 提出了如下猜想：若 $Delta^{{mon}}
(K_{n}^{c})色的Hamilton圈. 这个猜想至今还未被证明.我们研究了上述条件下的正常染色的路和圈.

Properly Colored Paths and Cycles in Complete Graphs

Let Kcn denote a complete graph on n vertices whose edges are colored in an arbitrary way.Let △mon(Kcn) denote the maximum number of edges of the same color incident with a vertex of Kcn.A properly colored cycle (path) in Kcn is a cycle (path) in which adjacent edges have distinct colors.B.Bollobás and P.Erd(o)s (1976) proposed the following conjecture:If △mon(Kcn) ＜ [n/2],then Kcn 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.