Chromatic roots and limits of dense graphs

TL;DR

This paper explores how recent findings on the continuity of chromatic polynomial roots extend to dense graph sequences, proposing new problems and conjectures in this area.

Contribution

It extends the theory of chromatic roots and their limits from sparse to dense graph sequences, introducing new questions and conjectures.

Findings

Extension of Benjamini--Schramm continuity results to dense graphs

Identification of new problems and conjectures in chromatic root theory

Insights into the behavior of chromatic roots in dense graph limits

Abstract

In this short note we observe that recent results of Abert and Hubai and of Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph sequences. We offer a number of problems and conjectures motivated by this observation.