An Approach to Hadwiger's Conjecture through Squares of Chordal Graphs

Hadwiger’s conjecture states that for every graph G, χ(G) ≤ η(G), where χ(G) is the chromatic number and η(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 F defined as follows: 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 χ(G), where each branch set is a path.