On the characterization of chordal graphs using Horn hypergeometric series

TL;DR

This paper provides an elementary combinatorial proof that characterizes chordal graphs through the Horn hypergeometric series representation of the inverse of their independence polynomials, building on prior algebraic and combinatorial connections.

Contribution

It offers a new, elementary proof of Radchenko and Villegas's characterization of chordal graphs using combinatorial methods, differing from previous algebraic approaches.

Findings

Chordal graphs are characterized by the inverse of their independence polynomials being Horn hypergeometric series.

The proof connects independence polynomials with multi-colored chromatic polynomials.

The approach simplifies understanding of the algebraic properties of chordal graphs.

Abstract

Radchenko and Villegas characterized the chordal graphs by the inverse of their independence polynomials being Horn hypergeometric series in Radchenko et al. in 2021. In this paper, we reprove their result using some elementary combinatorial methods. Our proof is different from their proof, and it is based on the connection between the inverse of the multi-variate independence polynomials and the multi-colored chromatic polynomials of graphs, established by Arunkumar et al. in 2018.