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Complete Minors, Independent Sets, and Chordal Graphs

By József Balogh, John Lenz and Hehui Wu


The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger’s Conjecture states that h(G) ≥ χ(G). Since χ(G)α(G) ≥ |V (G)|, Hadwiger’s Conjecture implies that α(G)h(G) ≥ |V (G)|. We show that (2α(G) − ⌈log τ(τα(G)/2)⌉)h(G) ≥ |V (G) | where τ ≈ 6.83. For graphs with α(G) ≥ 14, this improves on a recent result of Kawarabayashi and Song who showed (2α(G) − 2)h(G) ≥ |V (G) | when α(G) ≥ 3

Year: 2009
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