Complete Minors, Independent Sets, and Chordal Graphs

TL;DR

This paper establishes a new inequality relating the Hadwiger number, independence number, and number of vertices in a graph, improving bounds for graphs with large independence numbers.

Contribution

It introduces a novel inequality connecting Hadwiger number and independence number, refining previous bounds for graphs with large independence numbers.

Findings

Improves bounds on Hadwiger number for graphs with large independence number

Provides a new inequality involving independence number, Hadwiger number, and vertex count

Enhances understanding of graph minors in relation to independence and chromatic numbers

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.