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. (-∞, 0) and (0, 1) are maximal root-free intervals, as is (1, 32/27] (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).

chromatic polynomial | chromatic root | interval