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 four is bi-colored. The star chromatic index of $G$, denoted by ${\chi }_s^{\prime }(G)$, is the minimum $k$ such that $G$ admits a star edge coloring with $k$ colors. Bezegov{á} et al. (Star edge coloring of some classes of graphs, J. Graph Theory, 81(1), pp.73-82. 2016) conjectured that the star chromatic index of outerplanar graphs with maximum degree $\Delta$, is at most $\lfloor \frac{3\Delta }{2}\rfloor +1$. 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.