Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids

TL;DR

This paper establishes broad zero-free regions for multivariate Tutte polynomials of graphs and matroids, unifying and extending known results for chromatic and flow polynomials through simple, combinatorial proofs.

Contribution

It provides a general framework for zero-free regions of multivariate Tutte polynomials, simplifying proofs and explaining the significance of specific constants like 32/27.

Findings

Identifies zero-free regions for multivariate Tutte polynomials of graphs and matroids.

Unifies various known zero-free results as special cases.

Uses simple deletion-contraction and reduction techniques for proofs.

Abstract

The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zero-free regions of the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and employ deletion-contraction together with parallel and series reduction. In particular, they shed light on the origin of the curious number 32/27.