Independence number and clique minors

Since χ(G) · α(G) ≥ |V (G)|, Hadwiger’s Conjecture implies that any graph G has the complete graph Kdn α e as a minor, where n is the number of vertices of G and α is the maximum number of independent vertices in G. Motivated by this fact, it is shown that any graph on n vertices with independence number α ≥ 3 has the complete graph Kd n 2α−2 e as a minor. This improves the well-known theorem of Duchet and Meyniel and the recent improvement due to Kawarabayashi, Plummer, Toft. A new result on the odd version of Hadwiger’s Conjecure will also be mentioned.