Choosability conjectures and multicircuits

This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′ = ′ for every multigraph, is shown to imply that if a line graph is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t. It is proved that ch(H )= (H ) for many “small” graphs H , including in9ations of all circuits (connected 2-regular graphs) with length at most 11 except possibly length 9; and that ch′′(C)= ′′(C) (the total chromatic number) for various multicircuits C, mainly of even order, where a multicircuit is a multigraph whose underlying simple graph is a circuit. In consequence, it is shown that if any of the corresponding graphs H 2 or T (C) is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t. c © 2001 Elsevier Science B.V. All rights reserved.