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

Probably the best-known among the remaining unsolved problems in graph theory is Hadwiger’s Conjecture: If χ(G) denotes the chromatic number of graph G, then G has the complete graph Kχ(G) as a minor. The following would immediately follow from the truth of Hadwiger’s Conjecture: Conjecture: If G has n vertices and if α(G) denotes the independence number of G, then G has Kdn/α(G)e as a minor. In 1982, Duchet and Meyniel proved the following Theorem: If G has n vertices, then G has Kdn/(2α(G)−1)e as a minor. Until now, no improvement on the Duchet-Meyniel result has been obtained for graphs in general. In this talk, we present the following two results. Theorem 1: Every graph G on n vertices with α(G) ≥ 2 and clique number ω(G) has Kdn+ω(G))/(2α(G)−1)e as a minor. Theorem 2: Every graph G on n vertices with α(G) ≥ 3 has Kdn(1+c)/(2α(G)−1)e as a minor, for some constant c > 0, c depending on α(G).