On a new reformulation of Hadwigerʹs conjecture Original Research Article

Assuming that every proper minor closed class of graphs contains a maximum with respect to the homomorphism order, we prove that such a maximum must be homomorphically equivalent to a complete graph. This proves that Hadwigerʹs conjecture is equivalent to saying that every minor closed class of graphs contains a maximum with respect to homomorphism order. Let image be a finite set of 2-connected graphs, and let image be the class of graphs with no minor from image. We prove that if image has a maximum, then any maximum of image must be homomorphically equivalent to a complete graph. This is a special case of a conjecture of Nešetřil and Ossona de Mendez.