Hadwiger's conjecture forK6-free graphs

  title={Hadwiger's conjecture forK6-free graphs},
  author={Neil Robertson and Paul D. Seymour and Robin Thomas},
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… 

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