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 \(\Delta\), then \(\Delta+ 1 \leq \chi(G^2) \leq ch(G^2) \leq \Delta+2\) if \(\Delta \geq 3\), and \(ch(G^2) = \chi(G^2) = \Delta+1\) if \(\Delta \geq 6\). All inequalities here are sharp,even for outerplanar graphs.