Star edge coloring of generalized Petersen graphs

The star chromatic index of a graph $G$, denoted by ${\chi }_s^{\mathrm{\prime }}(G)$, is the smallest integer $k$ for which $G$ admits a proper edge coloring with $k$ colors such that every path and cycle of length four is not bicolored. Let $d$ be the greatest common divisor of $n$ and $k$. Zhu~et~al. (\footnotesize{Discussiones Mathematicae: Graph Theory, 41(2): 1265, 2021}) showed that for every integers $k$ and $n>2k$ with $d\ge 3$, generalized Petersen graph $GP(n,k)$ admits a 5-star edge coloring, with the exception of the case that $d=3$, $k\mathrm{\ne }d$ and $\frac{n}{3}=1(\mathrm{m}\mathrm{o}\mathrm{d}3)$. Also, they conjectured that for every $n>2k$, ${\chi }_s^{\mathrm{\prime }}(GP(n,k))\le 5$, except $GP(3,1)$. In this paper, we prove that for every $GP(n,k)$ with $n\ge 2k$ and $d\ge 3$ their conjecture is true. In fact, we provide a 5-star edge coloring of $GP(n,k)$, where $n\ge 2k$ and $d\ge 3$. We also obtain some results for 5-star edge coloring of $GP(n,k)$ with $d=2$. Moreover, Dvo{ř}{á}k et al. ({\footnotesize Journal of Graph Theory, 72(3):313-326, 2013}) conjectured that the star of chromatic index of subcubic graphs is at most 6. Thus, our results also prove this conjecture for the generalized Petersen graphs, as a class of subcubic graphs.