Improvements of the theorem of Duchet and Meyniel on Hadwiger's conjecture

Since chi(G) - alpha(G) >= n(G), Hadwiger's conjecture implies that any graph G has the complete graph K-[n/a] as a minor, where n = n(G) is the number of vertices of G and alpha = alpha(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 alpha-1)] as a minor. For alpha(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 alpha - 1) of Duchet and Meyniel can be improved to a larger constant, depending on alpha, for all alpha >= 3. Our method does not work for alpha = 2 and we only present some observations on this case. (C) 2005 Elsevier Inc. All rights reserved.