If x is an isolated vertex, then ? (x) ? 0 ,
Let x be a node If d t (x) ? 9 ,
Let x be a node ,
Let x be a node ,
This contradicts Lemma 30 and completes the proof of the theorem ,
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
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
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
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. ,
(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
The state of the three color problem In Quo vadis, graph theory?, pp.211-248, 1993. ,