Some new bounds on image-choosability Original Research Article

List T-colouring is a generalisation of list colouring in which the differences between adjacent colours must not lie in the set T. We present a conjecture giving an upper bound on the image-choosability image (where image) in terms of r and image which, if true, is tight for all values of r and image, and we prove the bound in the case image. We also prove the conjecture with the colouring number image in place of image, and use this result in conjunction with a theorem of Alon to establish an exponential upper bound on image in terms of r and image.