# Complete minors, independent sets, and chordal graphs

• Volume: 31, Issue: 4, page 639-674
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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József Balogh, John Lenz, and Hehui Wu. "Complete minors, independent sets, and chordal graphs." Discussiones Mathematicae Graph Theory 31.4 (2011): 639-674. <http://eudml.org/doc/270983>.

@article{JózsefBalogh2011,
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 (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.},
author = {József Balogh, John Lenz, Hehui Wu},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {clique minor; independence number; Hadwiger conjecture; chordal graphs},
language = {eng},
number = {4},
pages = {639-674},
title = {Complete minors, independent sets, and chordal graphs},
url = {http://eudml.org/doc/270983},
volume = {31},
year = {2011},
}

TY - JOUR
AU - József Balogh
AU - John Lenz
AU - Hehui Wu
TI - Complete minors, independent sets, and chordal graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 4
SP - 639
EP - 674
AB - 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.
LA - eng
KW - clique minor; independence number; Hadwiger conjecture; chordal graphs
UR - http://eudml.org/doc/270983
ER -

## References

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