中文摘要：

Opsut在1982年给出了任意图G的竞争数小于等于其边团覆盖数的结果。对于完全五部图KKn,n,n,n,n 当n≡1，5（rood6）时，本文首先构造一个极小的边团覆盖并从中得到其边团覆盖数，然后利用边团覆盖和竞争图之间的关系得到了其竞争数的一个新的上界，从而改进了由Opsut给出的完全五部图的上界。

英文摘要：

Opsut showed that, for any graph G, the competition number of G is less than or equal to the edge clique cover number of G in 1982. This paper comtruets the edge clique cover of minimum size and give the edge clique cover number for the complete pentapartite graphs Kn,n,n,n,n where n≡1,5（mod6）. Then the new bound for the competition number of K is given by the relation of the edge clique cover and the competition graph, which improve the result of Opsut on the bound for the competition number of K n,n,n,n,n