Expositio paper exp-20260807-9c0bc6
How Close Can a Chromatic Root Get to One?
In this paper we show that the chromatic function $P(G,t)$ is always a polynomial. We then ask a simple question: where does this polynomial equal zero? Tutte showed that two stretches of the real line, $(-\infty,0)$ and $(0,1)$, can never contain a root. Jac…
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Abstract
In this paper we show that the chromatic function $P(G,t)$ is always a polynomial. We then ask a simple question: where does this polynomial equal zero? Tutte showed that two stretches of the real line, $(-\infty,0)$ and $(0,1)$, can never contain a root. Jackson showed in 1993 that no root can lie between $1$ and $32/27$ either, and that this bound is sharp. We prove both of these results from scratch, along with all the tools needed to get there, and we close by looking at what has been discovered about this threshold since 1993.
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