Discussiones Mathematicae Graph Theory 31(4) (2011) 639-674
doi: 10.7151/dmgt.1571

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Complete Minors, Independent Sets, and Chordal Graphs

József Balogh

Department of Mathematics
University of Illinois
Urbana, IL 61801, USA
Department of Mathematics
University of California, San Diego
La Jolla, CA 92093, USA

John Lenz, Hehui Wu

Department of Mathematics
University of Illinois
Urbana, IL 61801, USA

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) ≥ χ(G). Since χ(G) α(G) ≥ | V(G) |, Hadwiger's Conjecture implies that α(G) h(G) ≥ | V(G) |. We show that (2 α(G) − ⌈ log τ (τ α(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 (2 α(G) −2) h(G) ≥ |V(G) | when α(G) ≥ 3.

Keywords: clique minor, independence number, Hadwiger conjecture, chordal graphs

2010 Mathematics Subject Classification: Primary: 05C83,
Secondary: 05C69, 05C70.

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Received 14 September 2009
Revised 15 September 2010
Accepted 6 October 2010