Linear bound in terms of maxmaxflow for the chromatic roots of series-parallel graphs

TL;DR

This paper establishes a linear bound on the chromatic roots of series-parallel graphs based on their maxmaxflow, extending to multivariate Tutte polynomials within a specific edge weight regime, nearly optimal.

Contribution

It introduces a new linear bound for chromatic roots of series-parallel graphs in terms of maxmaxflow, applicable to multivariate Tutte polynomials in the antiferromagnetic regime.

Findings

Chromatic roots lie within a specific disc determined by maxmaxflow.

The bound applies to complex roots, not just real.

The result is nearly sharp, within a factor of 1/log 2.

Abstract

We prove that the (real or complex) chromatic roots of a series-parallel graph with maxmaxflow Lambda lie in the disc |q-1| < (Lambda-1)/log 2. More generally, the same bound holds for the (real or complex) roots of the multivariate Tutte polynomial when the edge weights lie in the "real antiferromagnetic regime" -1 \le v_e \le 0. This result is within a factor 1/log 2 \approx 1.442695 of being sharp