List-colouring the square of an outerplanar graph Timothy J. Hetherington, Douglas R. Woodall * School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK Abstract It is proved that if G is a K2,3-minor-free graph with maximum degree Δ, then Δ + 1  χ(G 2 )  ch(G 2 )  Δ + 2 if Δ  3, and ch(G 2 )= χ(G 2 ) = Δ + 1 if Δ  6. All inequalities here are sharp, even for outerplanar graphs. Keywords: Choosability; Outerplanar graph; Minor-free graph; List square colouring 1 Introduction We use standard terminology, as defined in the references: for example [5] or [9]. The square G 2 of a graph G has the same vertex-set as G, and two vertices are adjacent in G 2 if they are within distance two of each other in G. There is great interest in discovering classes of graphs G for which the choosability or list chromatic number ch(G) is equal to the chromatic num- ber χ(G). The list-square-colouring conjecture (LSCC ) [5] is that, for every graph G, ch(G 2 )= χ(G 2 ). It is clear that this conjecture holds when the maximum degree Δ(G) of G is 0 or 1, and it can be deduced from the results of [7] when Δ(G) = 2: see [4]. In general, it is easy to see that Δ(G)+1  χ(G 2 )  ch(G 2 ). It is well known that a graph is outerplanar if and only if it is both K 4 -minor-free and K 2,3 -minor-free. Squares of K 4 -minor-free graphs were considered in [4]. For K 2,3 -minor-free graphs we have the following result, which is the same as for the slightly smaller class of outerplanar graphs. * Email: pmxtjh@nottingham.ac.uk, douglas.woodall@nottingham.ac.uk 1