On a new reformulation of Hadwiger's conjecture

13 years 4 months ago

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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 Nesetril and Ossona de Mendez.

Reza Naserasr, Yared Nigussie

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