Hadwiger’s Conjecture and Squares of Chordal 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. Since all split graphs are chordal, this implies that Hadwiger's conjecture is true for all graphs if and only if it is true for squares of chordal graphs. It is known that 2-trees are a class of chordal graphs. We further prove that Hadwiger's conjecture is true for squares of 2-trees. In fact, we prove the following stronger result: for any 2-tree T , its square T 2 has a clique minor of order χ(T 2 ) for which each branch set is a path, where χ(T 2 ) is the chromatic number of T 2 . As a corollary, we obtain that the same statement holds for squares of generalized 2-trees, and so Hadwiger's conjecture is true for such squares, where a generalized 2-tree is a graph constructed recursively by introducing a new vertex and making it adjacent to a single existing vertex or two adjacent existing vertices in each step, beginning with the complete graph of two vertices.