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 Nesetril and Ossona de Mendez.

• Type: Journal
• Year: 2006
• Where: DM
• Authors: Reza Naserasr, Yared Nigussie
• Keywords: Complete Graph, DM 2006, Homomorphism Order, Minor Closed Class