Combinatorica

, Volume 13, Issue 3, pp 279–361

• Neil Robertson
• Paul Seymour
• Robin Thomas
Article

## Abstract

In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph ont+1 vertices ist-colourable. Whent≤3 this is easy, and whent=4, Wagner's theorem of 1937 shows the conjecture to be equivalent to the four-colour conjecture (the 4CC). However, whent≥5 it has remained open. Here we show that whent=5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger's conjecture whent=5 is “apex”, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger's conjecture whent=5, because it implies that apex graphs are 5-colourable.

05 C 15 05 C 75

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## Authors and Affiliations

• Neil Robertson
• 1
• Paul Seymour
• 2
• Robin Thomas
• 3
1. 1.Dept. of MathematicsOhio State UniversityColumbusUSA
2. 2.MorristownUSA
3. 3.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA