A Basic Elementary Extension of the Duchet-Meyniel Theorem

TL;DR

This paper presents a fundamental extension of the Duchet-Meyniel theorem, aiming to simplify initial cases and facilitate further improvements, especially for graphs with independence number two, related to Hadwiger's conjecture.

Contribution

It introduces an elementary extension of the Duchet-Meyniel theorem to aid in advancing bounds for the Hadwiger number and independence number relationship.

Findings

Provides a simplified version of the Duchet-Meyniel inequality.

Facilitates future improvements in bounds for specific graph classes.

Highlights the open problem for independence number two cases.

Abstract

The Conjecture of Hadwiger implies that the Hadwiger number $h$ times the independence number $\alpha$ of a graph is at least the number of vertices $n$ of the graph. In 1982 Duchet and Meyniel proved a weak version of the inequality, replacing the independence number $\alpha$ by $2\alpha-1$, that is, $$(2\alpha-1)\cdot h \geq n.$$ In 2005 Kawarabayashi, Plummer and the second author published an improvement of the theorem, replacing $2\alpha - 1$ by $2\alpha - 3/2$ when $\alpha$ is at least 3. Since then a further improvement by Kawarabayashi and Song has been obtained, replacing $2\alpha - 1$ by $2\alpha - 2$ when $\alpha$ is at least 3. In this paper a basic elementary extension of the Theorem of Duchet and Meyniel is presented. This may be of help to avoid dealing with basic cases when looking for more substantial improvements. The main unsolved problem (due to Seymour) is to improve, even just slightly, the theorem of Duchet and Meyniel in the case when the independence number $\alpha$ is equal to 2. The case $\alpha = 2$ of Hadwiger's Conjecture was first pointed out by Mader as an interesting special case.