Degree sum conditions for graphs to have proper connection number 2
  A path $P$ in an edge-colored graph $G$ is a \emph{proper path} if no two adjacent edges of $P$ are colored with the same color. The graph $G$ is \emph{proper connected} if, between every pair of vertices, there exists a proper path in $G$. The \emph{proper connection number} $pc(G)$ of a connected graph $G$ is defined as the minimum number of colors to make $G$ proper connected. In this paper, we study the degree sum condition for a general graph or a bipartite graph to have proper connection number 2. First, we show that if $G$ is a connected noncomplete graph of order $n\geq 5$ such that $d(x)+d(y)\geq \frac{n}{2}$ for every pair of nonadjacent vertices $x,y\in V(G)$, then $pc(G)=2$ except for three small graphs on 6, 7 and 8 vertices. In addition, we obtain that if $G$ is a connected bipartite graph of order $n\geq 4$ such that $d(x)+d(y)\geq \frac{n+6}{4}$ for every pair of nonadjacent vertices $x,y\in V(G)$, then $pc(G)=2$. Examples are given to show that the above conditions are best possible. 
Hong Chang, Zhong Huang, Xueliang Li
math.CO