On a new reformulation of Hadwiger's conjecture

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 Hadwiger’s conjecture is equivalent to saying that every minor closed class of graphs contains a maximum with respect to homomorphism order. Let F be a finite set of 2-connected graphs, and let C be the class of graphs with no minor from F . We prove that if C has a maximum, then any maximum of C must be homomorphically equivalent to a complete graph. This is a special case of a conjecture of J. Nešetřil and P. Ossona de Mendez.