W. ?. Indeed and W. ?. , We first claim that v 1 = v 7 v 7 ) is a cycle C. It has no chord by Lemma ?? (ii), so C 2 = G 2 [S]. Moreover, p S (v i ) ? 4 if i is even and p S (v i ) ? 3 otherwise. C 2 is a cycle+triangle graph, thus, by Theorem ??, it is 3-choosable and so p S -choosable. This contradicts Lemma ??. Let w 1 (resp. w 7 ) be the neighbour of v 1 (resp. w 7 ) distinct from v 2 (resp. v 6 ) and for i ? {2, 4, 6}, let w i be the neighbour of v i not in {v i?1 , v i+1 }. Let W = {w 1, (ii), w 2 = v 4 and w 6 = v 4 and by Lemma ?? (iii), w 1 = v 4 and w 7 = v 4

O. V. Borodin, Criterion of chromaticity of a degree prescription (in Russian), Abstracts of IV All-Union Conf. on Theoretical Cybernetics (Novosibirsk), pp.127-128, 1977.

D. W. Cranston and S. Kim, List-coloring the square of a subcubic graph, Journal of Graph Theory, vol.17, issue.1, 2006.
DOI : 10.1002/jgt.20273

Z. Dvo?ák, R. Skrekovski, and M. Tancer, List-colouring squares of sparse subcubic graphs, 2005.

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.

H. Fleischner and M. Stiebitz, A solution to a Colouring problem of P. Erdös, Discrete Math, pp.39-48, 1992.

A. V. Kostochka and D. R. Woodall, Choosability conjectures and multicircuits, Discrete Mathematics, vol.240, issue.1-3, pp.123-143, 2001.
DOI : 10.1016/S0012-365X(00)00371-X

M. Montassier and A. Raspaud, A note on 2-facial coloring of plane graphs, Information Processing Letters, vol.98, issue.6, pp.211-266, 2006.
DOI : 10.1016/j.ipl.2006.02.013
URL : https://hal.archives-ouvertes.fr/hal-00307134

C. Thomassen, The square of a planar cubic graph is 7-colorable, manuscript, 2006.

G. Wegner, Graphs with given diameter and a coloring problem, 1977.