Chromatic Root-Free Interval

Author

Eric W. Weisstein
July 26, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/ChromaticRoot-FreeInterval.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ChromaticRoot-FreeInterval.html.

©2024 Wolfram Research, Inc. except for portions noted otherwise

Source

https://arxiv.org/pdf/2407.15860

Intervals

(-∞, 0)

With[{eps=1*^-20},Monitor[Select[GraphData[All],CountRoots[GraphData[gn=#,"ChromaticPolynomial"][x],{x,-Infinity,-eps}]>0&],gn]]//Quiet//Timing

DeleteCases[{#,AnyTrue[GraphData[#,"ChromaticRoots"],Function[r,NumericQ[r]&&Im[r]==0.&&r<0]]}&/@GraphData[All],{_,False}]//Timing

(0, 1)

With[{eps=1*^-20},Monitor[Select[GraphData[All],CountRoots[GraphData[gn=#,"ChromaticPolynomial"][x],{x,eps,1-eps}]>0&],gn]]//Quiet//Timing

DeleteCases[{#,AnyTrue[GraphData[#,"ChromaticRoots"],Function[r,NumericQ[r]&&Im[r]==0.&&0<r<1]]}&/@GraphData[All],{_,False}]//Timing

(1, (32/27)]

With[{eps=1*^-20},Monitor[Select[GraphData[All],CountRoots[GraphData[gn=#,"ChromaticPolynomial"][x],{x,1+eps,(32/27)}]>0&],gn]]//Quiet//Timing

DeleteCases[{#,AnyTrue[GraphData[#,"ChromaticRoots"],Function[r,NumericQ[r]&&Im[r]==0.&&1<r≤32/27]]}&/@GraphData[All],{_,False}]//Timing