In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t+1 vertices is t-colourable. When t≤3 this is easy, and when t=4, Wagner's theorem of 1937 shows the conjecture to be equivalent to the four-colour conjecture (the 4CC). However, when t≥5 it has remained open. Here we show that when t=5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger's conjecture when t=5 is "apex", that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger's conjecture when t=5, because it implies that apex graphs are 5-colourable. © 1993 Springer-Verlag.
CITATION STYLE
Robertson, N., Seymour, P., & Thomas, R. (1993). Hadwiger’s conjecture for K6-free graphs. Combinatorica, 13(3), 279–361. https://doi.org/10.1007/BF01202354
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