Advanced Search

An edge-colored graph $G$ is proper connected if every pair of vertices is connected by a proper path. The proper connection number of a connected graph $G$, denoted by $\mathrm{pc}\left(G\right)$, is the smallest number of colors that are needed to color the edges of $G$ in order to make it proper connected. In this paper, we obtain the sharp upper bound for $\mathrm{pc}\left(G\right)$ of a general bipartite graph $G$ and a series of extremal graphs. Additionally, we give a proper $2$-coloring for a connected bipartite graph $G$ having $\delta \left(G\right)\ge 2$ and a dominating cycle...