Circular choosability

We study circular choosability, a notion recently introduced by Mohar and by Zhu. First, we provide a negative answer to a question of Zhu about circular cliques. We next prove that cch(G) = O (ch(G) + ln |V (G)|) for every graph G. We investigate a generalisation of circular choosability, the circular f -choosability, where f is a function of the degrees. We also consider the circular choice number of planar graphs. Mohar asked for the value of τ := sup{cch(G) : G is planar}, and we prove that 6 ≤ τ ≤ 8, thereby providing a negative answer to another question of Mohar. We also study the circular choice number of planar and outerplanar graphs with prescribed girth, and graphs with bounded density. ∗This work was partially supported by Région Provence-Alpes-Côte d’Azur, and the European project ist fet Aeolus. ‡MASCOTTE, I3S (CNRS-UNSA) – INRIA, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France. E-mail: [email protected]. §School of Computer Science, McGill University, 3480 University Street, Montreal, H3A 2A7, Canada. The research in this paper was carried out while this author was a doctoral research student at the University of Oxford. He was partially supported by NSERC of Canada and the Commonwealth Scholarship Commission (UK). E-mail: [email protected]. ¶EURANDOM, Technische Universiteit Eindhoven, PO Box 513, 5600 M B Eindhoven, The Netherlands. The research in this paper was carried while this author was a doctoral research student at the University of Oxford. He was partially supported by EPSRC, The Oxford University Department of Statistics, Bekker-la-Bastide fonds and Prins Bernhard Cultuurfonds. E-mail: [email protected]. ‖Institute for Theoretical Computer Science (iti) and Department of Applied Mathematics (kam), Faculty of Mathematics and Physics, Charles University, Malostranské náměst́ı 25, 118 00 Prague 1, Czech Republic. The research in this paper was carried while this author was a doctoral research student in mascotte‡. E-mail: [email protected]. 1 in ria -0 04 96 43 2, v er si on 1 1 Ju l 2 01 0 Author manuscript, published in "Journal of Graph Theory 61, 4 (2009) 241--270" DOI : 10.1002/jgt.20375