An interval in which no chromatic root exists for any possible chromatic polynomial
is known as a chromatic root-free interval. An chromatic root-free interval that
cannot be extended is known as a maximal chromatic root-free interval.
and
are maximal root-free intervals, as is (Jackson 1993, Alikhani and Ghanbari 2024). Furthermore,
chromatic roots are dense in the complex
plane (Sokal 2004, Cameron and Morgan 2016). The plots above show a histogram
of chromatic roots along the real axis and the positions
of chromatic roots in the complex plane for graphs
in GraphData
(the latter of which shows clear deviations from uniformity).
Alikhani, S. and Ghanbari, N. "Golden Ratio in Graph Theory: A Survey." 9 Jul 2024. https://arxiv.org/abs/2407.15860.Cameron,
P. J. and Morgan, K. "Algebraic Properties of Chromatic Roots." 3
Oct 2016. https://arxiv.org/abs/1610.00424.Dong,
F. M., Koh, K. M.; and Teo, K. L. Chromatic
Polynomials and Chromaticity of Graphs. Singapore: World Scientific, 2005.Jackson,
B. "A Zero Free Interval for Chromatic Polynomials of Graphs." Combin.
Probab. Comput.2, 325-336, 1993.Sokal, A. D. "Chromatic
Roots Are Dense in the Whole Complex Plane." Combin. Probab. and Comput.13,
221-261, 2004.
and 
are maximal root-free intervals, as is ![(1,32/27]](/images/equations/ChromaticRoot-FreeInterval/Inline3.svg)
(Jackson 1993, Alikhani and Ghanbari 2024). Furthermore,
chromatic roots are dense in the complex
plane (Sokal 2004, Cameron and Morgan 2016). The plots above show a histogram
of chromatic roots along the real axis and the positions
of chromatic roots in the complex plane for graphs
in GraphData
(the latter of which shows clear deviations from uniformity).
