A directed path in a digraph is proper if any two consecutive arcs on the path have distinct colors. An arc-colored digraph D is proper connected if for any two distinct vertices x and y of D, there are both proper $(x,y)$-directed paths and proper $(y,x)$-directed paths in D. The proper connection number $\overrightarrow{pc}(D)$ of a digraph D is the minimum number of colors that can be used to make D proper connected. Obviously, if a digraph has a proper connection number, it must be strongly connected, and $\overrightarrow{pc}(D)=1$ if and only if D is complete. Magnant et al. showed that $\overrightarrow{pc}(D)\le 3$ for all strong digraphs D, and Ducoffe et al. proved that deciding whether a given digraph has proper connection number at most two is NP-complete. In this paper, we give a few classes of strong digraphs with proper connection number two, and from our proofs one can construct an optimal arc-coloring for a digraph of order n in time $O(n^3)$.

如果路径上的任何两个连续弧具有不同的颜色，则有向图中的有向路径是正确的。的圆弧各色有向图d被适当地连接，如果对任意两个不同的顶点X和ÿ的d，有两个合适的$（X，ÿ）$导向的路径和适当的 $（ÿ，X）$D中的定向路径。正确的连接号$\overrightarrow{pC}（d）$图D的“ D”是可用于使D正确连接的最小颜色数。显然，如果有向图的连接号正确，则必须将其牢固连接，并且$\overrightarrow{pC}（d）=\mathrm{1个}$当且仅当D完成时。Magnant等。表明$\overrightarrow{pC}（d）\le 3$对于所有强有向图D和Ducoffe等。证明确定给定的有向图最多具有两个正确的连接数是NP完全的。在本文中，我们给予适当的连接数两强有向图的几类，从我们的证明可以构造一个最佳的圆弧着色订单的有向图ñ的时间$Ø（{ñ}^3）$。