Chromatic polynomials of complements of bipartite graphs

TL;DR

This paper investigates the chromatic polynomials of bicliques, providing a formula, conditions for polynomial equivalence, and proving a case of the alpha+n conjecture related to chromatic roots.

Contribution

It derives a general formula for biclique chromatic polynomials and proves the cubic case of the alpha+n conjecture using a specific biclique subfamily.

Findings

Derived a formula for the chromatic polynomial of any biclique.

Established conditions for when two bicliques have the same splitting field.

Proved the cubic case of the alpha+n conjecture for chromatic roots.

Abstract

Bicliques are complements of bipartite graphs; as such each consists of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic polynomial of an arbitrary biclique, and use this to give certain conditions under which two of the graphs have chromatic polynomials with the same splitting field. Finally, we use a subfamily of bicliques to prove the cubic case of the $\alpha +n$ conjecture, by showing that for any cubic integer $\alpha$, there is a natural number $n$ such that $\alpha +n$ is a chromatic root.