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... is still open for k ≥ 7. Let G be a graph with n vertices, chromatic number k, and independence number α. In a proper k-coloring of G, each of the k independent sets has cardinality at most α, so k ≥ n/α. Hadwiger’s conjecture therefore implies that every graph with n vertices and independence number α has a clique minor of order at least n/α. In 1982, Duchet and Meyniel [8] proved that this bound holds within a factor 2. More precisely, they showed that every graph with n vertices and independence number α contains a clique minor of order at least n2α−1 . There has been several improvements [16, 26, 18, 12, 13, 17, 2] on the bound of Duchet and Meyniel. Kawarabayashi, Plummer, and Toft [12] showed that every graph with n vertices and independence number α ≥ 3 contains a clique minor of order at least n2α−3/2 , which was later improved to n 2α−2 by Kawarabayashi and Song [13]. Wood [25] improves on the bound of Kawarabayashi, Plummer, and Toft on the order of a largest clique minor by an additive constant. Very recently, Balogh, Lenz, and Wu [2] improved these bounds to n2α−O(logα) . Our main result gives the first improvement of these bounds by an absolute constant factor. Theorem 1. Let c = 29− √ 813 28 >...

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...lt by Chudnovsky and Ovetsky Fradkin [36] on large complete minors in claw-free graphs, that every non-empty graph G contains a complete minor of order larger than .504n(G)/α(G). Balogh, Lenz, and Wu =-=[12]-=- obtained a better lower bound for all graphs with independence number at most 230. The Conjecture of Hadwiger implies that the Hadwiger number η times the indeAbstract pendence number α of a graph is...

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...to a constant factor. More specifically, in 1982, Duchet and Meyniel [9] proved that a graph on n vertices 1 and stability number α has a clique minor of size n 2α−1 . There have been several results =-=[12, 13, 20, 3]-=- making improvements on the factor 2α− 1 but none of them improving on the constant factor of 1 2 . Recently, Fox showed that the main result of this paper can be used to make the first improvement on...

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...ter is joint work with József Balogh and Hehui Wu. It was accepted for publication in Discussiones Mathematicae Graph Theory under the title Complete Minors, Independent Sets, and Chordal Graphs, see =-=[8]-=- 47Theorem 50. Let G be an n-vertex graph. Then K ⌈n/r⌉ is a minor of G, where r = 2α(G) − ⌈log τ (τα(G)/2)⌉ and τ = 2√ 2 √ 2 − 1 ≈ 6.83. Using a more careful analysis, the bound for α(G) = 5 can be ...

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...direction by Duchet and Meyniel [5]. Part (b), which is strong when t is small, is by the author [23]. Part (c), which builds upon a similar result by Fox [6], is due to Balogh and Kostochka [2]. See =-=[3, 9, 10, 14, 15]-=- for related results. Lemma 2. Every Kt+1-minor-free graph on n vertices has an independent set of size α, where (a) (2α− 1)t ≥ n , (b) (2α− 1)(2t − 5) ≥ 2n− 5 for t ≥ 5 , (c) (2− γ)αt ≥ n , where γ =...