Any 7-Chromatic Graphs Has K 7 Or K 4,4 As A Minor

In 1943, Hadwiger made the conjecture that every k-chromatic graph has a K k -minor. This conjecture is, perhaps, the most interesting conjecture of all graph theory. It is well known that the case k=5 is equivalent to the Four Colour Theorem, as proved by Wagner [39] in 1937. About 60 years later, Robertson, Seymour and Thomas [29] proved that the case k=6 is also equivalent to the Four Colour Theorem. So far, the cases k≥7 are still open and we have little hope to verify even the case k=7 up to now. In fact, there are only a few theorems concerning 7-chromatic graphs, e. g. [17].

In this paper, we prove the deep result stated in the title, without using the Four Colour Theorem [1,2,28]. This result verifies the first unsettled case m=6 of the (m,1)-Minor Conjecture which is a weaker form of Hadwiger’s Conjecture and a special case of a more general conjecture of Chartrand et al. [8] in 1971 and Woodall [42] in 1990.

The proof is somewhat long and uses earlier deep results and methods of Jørgensen [20], Mader [23], and Robertson, Seymour and Thomas [29].

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Correspondence to Ken-ichi Kawarabayashi*.

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Dedicated to Professor Mike Plummer on the occasion of his sixty-fifth birthday.

* Research partly supported by the Japan Society for the Promotion of Science for Young Scientists.

† Research partly supported by the Danish Natural Science Research Council.

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Kawarabayashi*, Ki., Toft†, B. Any 7-Chromatic Graphs Has K 7 Or K 4,4 As A Minor. Combinatorica 25, 327–353 (2005).

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Mathematics Subject Classification (2000):

  • 05C15
  • 05C83