Hadwiger's Conjecture for squares of 2-Trees

TL;DR

This paper proves Hadwiger's conjecture for squares of 2-trees, a specific subclass of graphs, by demonstrating that their squares contain clique minors matching their chromatic number.

Contribution

It establishes the conjecture for squares of 2-trees and shows the equivalence of the conjecture for all graphs to that for squares of split graphs.

Findings

Hadwiger's conjecture holds for squares of 2-trees.

Clique minors in squares of 2-trees match their chromatic number.

The conjecture for all graphs reduces to split graphs.

Abstract

Hadwiger's conjecture asserts that any graph contains a clique minor with order no less than the chromatic number of the graph. We prove that this well-known conjecture is true for all graphs if and only if it is true for squares of split graphs. This observation implies that Hadwiger's conjecture for squares of chordal graphs is as difficult as the general case, since chordal graphs are a superclass of split graphs. Then we consider 2-trees which are a subclass of each of planar graphs, 2-degenerate graphs and chordal graphs. We prove that Hadwiger's conjecture is true for squares of $2$-trees. We achieve this by proving the following stronger result: for any $2$-tree $T$, its square $T^2$ has a clique minor of order $\chi(T^2)$ for which each branch set induces a path, where $\chi(T^2)$ is the chromatic number of $T^2$.