Complete Minors, Independent Sets, and Chordal Graphs

Details Complete Minors, Independent Sets, and Chordal Graphs Complete Minors, Independent Sets, and Chordal Graphs

2009-07-14

arxiv.org/abs/0907.2421v2

Discuss. Math. Graph Theory. vol 31(4). 2011. 639-674

Mathematics

Combinatorics

Scientific paper

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.

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