A small non-image-colorable planar graph

Let A be an abelian group. The graph G is A-colorable if for every orientation image of G and for every image, there is a vertex-coloring image such that image for each image. This notion was introduced by Jaeger et al. [Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165–182]. In this note, we give a smaller example of planar graph, with 18 vertices, which is not image-colorable. The earlier example due to Král et al. [A note on group coloring, J. Graph Theory 50 (2005) 123–129] has 34 vertices.