On the location of chromatic zeros of series-parallel graphs

04/21/2022

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by   Ferenc Bencs, et al.

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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 {z∈ℂ| |z-1| ≤ 1}, we show density of these zeros in the half plane (q)>3/2 and we show there exists an open region U containing the interval (0,32/27) such that U∖{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 Δ there exists a series-parallel graph for which all vertices but one have degree at most Δ and whose chromatic polynomial has a zero with real part exceeding Δ.