Let G be a graph with n vertices and independence number α. Hadwiger’s conjecture implies that G contains a clique minor of order at least n/α. In 1982, Duchet and Meyniel proved that this bound holds within a factor 2. Our main result gives the first improvement on their bound by an absolute constant factor. We show that G contains a clique minor of order larger than .504n/α. We also prove rel...

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 o...

A graph on n vertices is called pancyclic if it contains a cycle of length ` for all 3 ≤ ` ≤ n. In 1972, Erdős proved that if G is a Hamiltonian graph on n > 4k vertices with independence number k, then G is pancyclic. He then suggested that n = Ω(k) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such th...