Degree sums and proper connection number of graphs

An edge-colored graph is called if any two vertices are connected by a properly colored path. The of a graph , denoted by , is the smallest number of colors that are needed to color such that it is properly connected. A separating vertex of is if it is contained in at least three blocks, each of which contains at most three vertices. Let be a graph of order without bad vertices. In this paper, we show that if the degree sums of any three nonadjacent vertices of a graph is at least and , then , which generalizes the results due to Borozan et al. (2012) and Chang et al. (2019).