F. Havet, A. D. King, and @. Subcase, Since L(u k ) ? 2q + 1, there exist two colours a and b in L(u k ) \ {c 1 } such that [a] q ? [b] q = / 0. We define the list assignment L ? of G ? by L ? (v) := L(v) if v / ? {v 1(v) \ {a, b} otherwise, Again |L ?, vol.2, issue.2

|. ?. Then, ? 2q + 3 ? 2 = 2q + 1 for each i = j. Thus we can apply the induction hypothesis to G ? and L ? . Now we complete the colouring of G by colouring u k with a if c

|. ?. Then, ? 2q + 3 ? 2 = 2q + 1 for each i. Thus we can apply the induction hypothesis to G ? and L ? . Now we complete the colouring of G by colouring u k with a if c

M. Tarsi, Degrees and choice numbers Random Structures Algorithms Colorings and orientations of graphs, Alon, C. McDiarmid, and B. Reed. Star arboricity. Combinatorica Combinatorica, vol.16, issue.122, pp.364-368375, 1992.

H. Broersma, F. V. Fomin, P. A. Golovach, and G. J. Woeginger, Backbone Colorings for Networks, Proceedings of the 29th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2003), pp.2880131-142, 2003.
DOI : 10.1007/978-3-540-39890-5_12

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.7.3880

H. Broersma, F. V. Fomin, P. A. Golovach, and G. J. Woeginger, Backbone colorings for graphs: Tree and path backbones, Journal of Graph Theory, vol.94, issue.2, pp.137-152, 2007.
DOI : 10.1002/jgt.20228

H. J. Broersma, J. Fujisawa, L. Marchal, D. Paulusma, A. N. Salman et al., <mml:math altimg="si11.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>??</mml:mi></mml:math>-backbone colorings along pairwise disjoint stars and matchings, Discrete Mathematics, vol.309, issue.18, pp.5596-5609, 2009.
DOI : 10.1016/j.disc.2008.04.007

P. Erd?-os, A. L. Rubin, and H. Taylor, Choosability in graphs, Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing Congress. Numer., XXVI, pp.125-157, 1979.

R. Evans and P. Montgomery, 6631, The American Mathematical Monthly, vol.98, issue.9, pp.870-872, 1991.
DOI : 10.2307/2324290

S. L. Hakimi, On the degrees of the vertices of a directed graph, Journal of the Franklin Institute, vol.279, issue.4, pp.290-308, 1965.
DOI : 10.1016/0016-0032(65)90340-6

F. Havet, R. Kang, T. Müller, and J. Sereni, Circular choosability, Journal of Graph Theory, vol.48, issue.3, pp.241-334, 2009.
DOI : 10.1002/jgt.20375

URL : https://hal.archives-ouvertes.fr/inria-00086981

F. Havet, A. D. King, M. Liedloff, and I. Todinca, (Circular) backbone coloring: tree backbones in planar graphs, INRIA Research Report, vol.8512, 2012.

C. Mcdiarmid, On the span in channel assignment problems: bounds, computing and counting. Discrete Math, pp.387-397, 2003.

]. B. Mohar, Choosability for the circular chromatic number, 2003.

S. Norine, T. Wong, and X. Zhu, Circular choosability via combinatorial Nullstellensatz, Journal of Graph Theory, vol.48, issue.3, pp.190-204, 2008.
DOI : 10.1002/jgt.20333

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.183.5185

M. Tarsi, On the decomposition of a graph into stars, Discrete Mathematics, vol.36, issue.3, pp.299-304, 1981.
DOI : 10.1016/S0012-365X(81)80025-8

C. Thomassen, Every Planar Graph Is 5-Choosable, Journal of Combinatorial Theory, Series B, vol.62, issue.1, pp.180-181, 1994.
DOI : 10.1006/jctb.1994.1062

URL : http://doi.org/10.1006/jctb.1994.1062

X. Zhu, Circular choosability of graphs, Journal of Graph Theory, vol.8, issue.3, pp.210-218, 2005.
DOI : 10.1002/jgt.20051