The Edge Choosability of $C_n\times P_n$

Abstract

Let $G_{n,m}=C_n\times P_m$, be the cartesian product of an $n$-cycle $C_n$ and a path $P_m$ of length $m-1$. We prove that ${\chi }^{\prime }(G_{n,m})={\chi }^{\prime }(G_{n,m})=4$ if $m\ge 3$, which implies that the list-edge-coloring conjecture (LLECC) holds for all graphs $G_{n,m}$.