On a new reformulation of Hadwiger's conjecture

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Abstract

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 Nešetřil and Ossona de Mendez. © 2006 Elsevier B.V. All rights reserved.

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Naserasr, R., & Nigussie, Y. (2006). On a new reformulation of Hadwiger’s conjecture. Discrete Mathematics, 306(23 SPEC. ISS.), 3136–3139. https://doi.org/10.1016/j.disc.2005.06.043

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