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Keywords:
proper coloring; proper connection number; bipartite graph; dominating set
Summary:
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 ${\rm pc}(G)$, 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 ${\rm pc}(G)$ 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 $\delta (G)\geq 2$ and a dominating cycle or a dominating complete bipartite subgraph, which implies ${\rm pc}(G)=2$. Furthermore, we get that the proper connection number of connected bipartite graphs with $\delta \geq 2$ and ${\rm diam}(G)\leq 4$ is two.
References:
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