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].