Minimum degree and size conditions for the graphs of proper connection number 2

Abstract

An edge-colored graph $G$ is called properly connected if every pair of distinct vertices of $G$ is connected by a proper path. For a connected graph $G$, the proper connection number of $G$, denoted by $pc\left(G\right)$, is defined as the smallest number of colors that are needed in order to make $G$ properly connected. Guan, Xue, Cheng and Yang (Guan et al., 2019) conjectured that if $G$ is a connected graph of order $n$, $\delta \ge 3$ and $\left|E\left(G\right)\right|\ge \left(\frac{n-p-(3-p)(\delta +1)}{2}\right)+(3-p)\left(\frac{\delta +1}{2}\right)+4$, then $pc\left(G\right)\le 2$, where $p$ takes the value 1 if $\delta =3$ and 0 if $\delta \ge 4$. We confirm the validity of the conjecture.