Improvements of the theorem of Duchet and Meyniel on Hadwigerʹs conjecture

Since χ ( G ) · α ( G ) ⩾ n ( G ) , Hadwigerʹs conjecture implies that any graph G has the complete graph K ⌈ n / α ⌉ as a minor, where n = n ( G ) is the number of vertices of G and α = α ( G ) is the maximum number of independent vertices in G. Duchet and Meyniel [Ann. Discrete Math. 13 (1982) 71–74] proved that any G has K ⌈ n / ( 2 α - 1 ) ⌉ as a minor. For α ( G ) = 2 G has K ⌈ n / 3 ⌉ as a minor. Paul Seymour asked if it is possible to obtain a larger constant than 1 3 for this case. To our knowledge this has not yet been achieved. Our main goal here is to show that the constant 1 / ( 2 α - 1 ) of Duchet and Meyniel can be improved to a larger constant, depending on α , for all α ⩾ 3 . Our method does not work for α = 2 and we only present some observations on this case.