. Lemma-34, If x is an isolated vertex, then ? (x) ? 0

. Lemma-36, Let x be a node If d t (x) ? 9

. Lemma-38, Let x be a node

. Lemma-39, Let x be a node

?. Thus, This contradicts Lemma 30 and completes the proof of the theorem

K. Appel and W. Haken, Every planar map is four colorable, Bulletin of Americain mathematical society, vol.82, 1976.
DOI : 10.1090/s0002-9904-1976-14122-5

URL : http://www.ams.org/bull/1976-82-05/S0002-9904-1976-14122-5/S0002-9904-1976-14122-5.pdf

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

URL : http://www.ii.uib.no/%7Efomin/articles/2007/2007c.pdf

V. Campos, F. Havet, R. Sampaio, and A. Silva, Backbone colouring: Tree backbones with small diameter in planar graphs, Theoretical Computer Science, vol.487, pp.0-50, 2013.
DOI : 10.1016/j.tcs.2013.03.003

URL : https://hal.archives-ouvertes.fr/hal-00758548

H. Grötzsch, Zur Theorie der diskreten Gebilde. VII. Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe, vol.8, pp.109-120, 1958.

F. Havet, A. D. King, M. Liedloff, and I. Todinca, (Circular) backbone colouring: Forest backbones in planar graphs, Discrete Applied Mathematics, vol.169, pp.119-134, 2014.
DOI : 10.1016/j.dam.2014.01.011

URL : https://hal.archives-ouvertes.fr/hal-00957243

R. Steinberg, The state of the three color problem In Quo vadis, graph theory?, pp.211-248, 1993.