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

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