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arXiv · 2204.10038

On the location of chromatic zeros of series-parallel graphs

Ferenc Bencs·Jeroen Huijben·Guus Regts
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Abstract

In this paper we consider the zeros of the chromatic polynomial of series-parallel graphs. Complementing a result of Sokal, showing density outside the disk $|q-1|\leq1$, we show density of these zeros in the half plane $\Re(q)>3/2$ and we show there exists an open region $U$ containing the interval $(0,32/27)$ such that $U\setminus\{1\}$ does not contain zeros of the chromatic polynomial of series-parallel graphs. We also disprove a conjecture of Sokal by showing that for each large enough integer $\Delta$ there exists a series-parallel graph for which all vertices but one have degree at most $\Delta$ and whose chromatic polynomial has a zero with real part exceeding $\Delta$.

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Ferenc Bencs, Jeroen Huijben, Guus Regts. 2022-04-21. On the location of chromatic zeros of series-parallel graphs. https://arxiv.org/abs/2204.10038

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