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    $\begingroup$ Is it possible that $\chi_G(x+4)$ has positive coefficients? I looked through the table of chromatic polys. of triangulations in Birkhoff-Lewis up to 19 vertices, and found no counterexamples. $\endgroup$ Commented Jun 16, 2015 at 13:43
  • $\begingroup$ Ian: This paper claims real roots arbitrarily close to 4: link.springer.com/article/10.1007/… $\endgroup$ Commented Apr 30, 2016 at 6:04
  • $\begingroup$ "and it's well known that the roots can't be real numbers greater than or equal to 5." - I guess you rather mean "and it's well known that the roots can't be natural numbers greater than or equal to 5." $\endgroup$ Commented Mar 20, 2024 at 23:31
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    $\begingroup$ @Nubok: No, I meant real numbers. See this blog post and Greg Kuperberg's comment. There's a skein theory argument where you can rewrite the chromatic polynomial for small faces as a sum of simpler terms with positive coefficients so long as x > 5. $\endgroup$ Commented Mar 21, 2024 at 19:10