Hadwiger's conjecture

Abstract

Hadwiger's conjecture states that any graph that does not have the complete graph K k as a minor is ( k − 1)- colourable. It is well known that the case k = 5 is equivalent to the four-colour theorem. In 1993 Robertson, Seymour and Thomas proved that the case k = 6 is also equivalent to the four-colour theorem. For k ≥ 7, the conjecture is still open. Our main focus in this chapter is to present recent results related to minimal counter-examples to Hadwiger's conjecture and some variations. We also consider algorithmic aspects of the conjecture and some of its variants, including the list-colouring version (which is false), the odd case of the conjecture, Hajos's conjecture and totally odd subdivisions, and Hadwiger's conjecture for some special classes of graphs . Introduction This chapter is motivated by Hadwiger's conjecture from 1943, which suggests a farreaching generalization of the four-colour theorem. It is among the most challenging open problems in all of graph theory. Hadwiger's conjecture (strong version) For all k , every k -colourable graph contains the complete graph K k as a minor. For k ≤ 3, Hadwiger's conjecture is easy to prove, and for k = 4, it was proved independently by Hadwiger himself [31] and Dirac [21]. For k = 5, however, it becomes extremely difficult. In 1937 Wagner [74] proved that this case is equivalent to the four-colour theorem, so, given that result in [2], [3], [56], it follows that the case k = 5 also holds. In the deepest theorem in this area so far, Robertson, Seymour and Thomas [62] proved in 1993 that a minimal counter-example to the case k = 6 must have a vertex whose removal leaves a planar graph, so this case too follows from the four-colour theorem. For k ≥ 7, the conjecture is still open. For k = 7, Kawarabayashi and Toft [46] proved that every 7-chromatic graph has K 7 or K 4,4 as a minor, and recently Kawarabayashi [38] proved that every 7-chromatic graph has K 7 or K 3,5 as a minor.