The optimal proper connection number of a graph with given independence number

An edge-colored connected graph $G$ is properly connected if between every pair of distinct vertices, there exists a path such that no two adjacent edges have the same color. Fujita (2019) introduced the optimal proper connection number pc_opt(G) for a monochromatic connected graph $G$, to make a connected graph properly connected efficiently. More precisely, pc_opt ($G$) is the smallest integer $p+q$ when one converts a given monochromatic graph $G$ into a properly connected graph by recoloring $p$ edges with $q$ colors. In this paper, we show that pc_opt ($G$) has an upper bound in terms of the independence number $\alpha \left(G\right)$. Namely, we prove that for a connected graph $G$, pc_opt ($G$)$\le \frac{5\alpha \left(G\right)-1}{2}$. Moreover, for the case $\alpha \left(G\right)\le 3$, we improve the upper bound to 4, which is tight.