Degree sums and proper connection number of graphs.

An edge-colored graph G is called properly connected if any two vertices are connected by a properly colored path. The proper connection number of a graph G , denoted by p c (G) , is the smallest number of colors that are needed to color G such that it is properly connected. A separating vertex of G is bad if it is contained in at least three blocks, each of which contains at most three vertices. Let G be a graph of order n without bad vertices. In this paper, we show that if the degree sums of any three nonadjacent vertices of a graph G is at least 3 n / 4 and n > 48 , then p c (G) = 2 , which generalizes the results due to Borozan et al. (2012) and Chang et al. (2019). [ABSTRACT FROM AUTHOR]