Improved bounds for zeros of the chromatic polynomial on bounded degree graphs

Abstract

We prove that for any graph $G$ of maximum degree at most $\Delta$, the zeros of its chromatic polynomial ${\chi }_G(z)$ (in $\mathbb{C}$) lie outside the disk of radius $5.02\Delta$ centered at $0$. This improves on the previously best known bound of approximately $6.91\Delta$. In the case of graphs of high girth we can improve this. We prove that for every $g$ there is a constant $K_g$ such that for any graph $G$ of maximum degree at most $\Delta$ and girth at least $g$, the zeros of its chromatic polynomial ${\chi }_G(z)$ lie outside the disk of radius $K_g\Delta$ centered at $0$ where $K_g\to 1+e\approx 3.72$ as $g\to \infty$. Finally, we give improved bounds on the Fisher zeros of the partition function of the Ising model.

Eq (2.4) is not correct and as such this invalidates Theorem 2.3 and consequently all the claimed results on the modulus of the zeros of chromatic polynomial. As fas as we can tell the results for the edge based block polynomials are correct (this concerns Sections 4 and 5). We will probably resubmit this part as part of a new paper at some point in the future

References in corpus (2)

• Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials
• Fisher zeros and correlation decay in the Ising model