Complete Minors, Independent Sets, and Chordal Graphs
Abstract
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) >= (G). Since (G) α(G) >= |V(G)|, Hadwiger's Conjecture implies that α(G) h(G) >= |V(G)|. We show that (2 α(G) - logt(t α(G)/2) ) h(G) ≥ |V(G)| where t is approximately 6.83. For graphs with α(G) ≥ 14, this improves on a recent result of Kawarabayashi and Song who showed (2 α(G) - 2) h(G) >= |V(G)| when α(G) >= 3.
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