Star Edge Coloring of Cactus Graphs

Abstract

A star edge coloring of a graph G is a proper edge coloring of G such that no path or cycle of length 4 is bicolored. The star chromatic index of G, denoted by $\chi ^{\prime }_{s}(G)$, is the minimum k such that G admits a star edge coloring with k colors. Bezegová et al. (J Graph Theory 81(1):73–82, 2016) conjectured that the star chromatic index of outerplanar graphs with maximum degree $\Delta$ is at most $\left\lfloor \frac{3\Delta }{2}\right\rfloor +1$. In this paper, we prove this conjecture for a class of outerplanar graphs, namely Cactus graphs, wherein every edge belongs to at most one cycle.