Injective Edge Chromatic Index of Generalized Petersen Graphs

Abstract

An injective k-edge coloring of a graph $G=(V(G),E(G))$ is a k-edge coloring $\varphi$ of G such that $\varphi (e_1)\ne \varphi (e_3)$ for any three consecutive edges $e_1,e_2$ and $e_3$ of a path or a 3-cycle. The injective edge chromatic index of G, denoted by $\chi _i'(G)$, is the minimum k such that G has an injective k-edge coloring. In this paper, we consider the injective edge coloring of the generalized Petersen graph P(n, k). We show that $\chi _i'(P(n,k))\le 4$ if $n\equiv 0(mod~4)$ and $k\equiv 1(mod~2)$; and $\chi _i'(P(n,k))\le 5$ if $n\equiv 2(mod~4)$ and $k\equiv 1(mod~2)$. Moreover, $\chi _i'(P(n,3))\le 5$, $\chi _i'(P(2k+1,k))\le 5$ and $\chi _i'(P(2k+2,k))\le 5$.