Complete Minors, Independent Sets, and Chordal Graphs
Combinatorics
2011-10-14 v2
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) >= \chi(G). Since \chi(G) \alpha(G) >= |V(G)|, Hadwiger's Conjecture implies that \alpha(G) h(G) >= |V(G)|. We show that (2 \alpha(G) - \lceil log_t(t \alpha(G)/2) \rceil) h(G) \geq |V(G)| where t is approximately 6.83. For graphs with \alpha(G) \geq 14, this improves on a recent result of Kawarabayashi and Song who showed (2 \alpha(G) - 2) h(G) >= |V(G)| when \alpha(G) >= 3.
Keywords
Cite
@article{arxiv.0907.2421,
title = {Complete Minors, Independent Sets, and Chordal Graphs},
author = {Jozsef Balogh and John Lenz and Hehui Wu},
journal= {arXiv preprint arXiv:0907.2421},
year = {2011}
}