Hadwiger’s Conjecture and Squares of Chordal Graphs

Abstract

Hadwiger’s conjecture states that for every graph G, $\chi (G)\le \eta (G)$, where $\chi (G)$ is the chromatic number and $\eta (G)$ is the size of the largest clique minor in G. In this work, we show that to prove Hadwiger’s conjecture in general, it is sufficient to prove Hadwiger’s conjecture for the class of graphs $\mathcal {F}$ defined as follows: $\mathcal {F}$ is the set of all graphs that can be expressed as the square graph of a split graph. Since split graphs are a subclass of chordal graphs, it is interesting to study Hadwiger’s Conjecture in the square graphs of subclasses of chordal graphs. Here, we study a simple subclass of chordal graphs, namely 2-trees and prove Hadwiger’s Conjecture for the squares of the same. In fact, we show the following stronger result: If G is the square of a 2-tree, then G has a clique minor of size $\chi (G)$, where each branch set is a path.

L. Sunil Chandran—Part of the work was done when this author was visiting Max Planck Institute for Informatics, Saarbruecken, Germany supported by Alexander von Humboldt Fellowship.