List-colouring the Square of an Outerplanar Graph

Abstract

It is proved that if $G$ is a $K_{2,3}$-minor-free graph with maximum degree $\mathrm{\Delta }$, then $\mathrm{\Delta }+1\le \chi (G^2)\le ch(G^2)\le \mathrm{\Delta }+2$ if $\mathrm{\Delta }\ge 3$, and $ch(G^2)=\chi (G^2)=\mathrm{\Delta }+1$ if $\mathrm{\Delta }\ge 6$. All inequalities here are sharp,even for outerplanar graphs.