LetP(λ) be the chromatic polynomial of a graph. We show thatP(5)-1P(6)2P(7)-1can be arbitrarily small, disproving a conjecture of Welsh (and of Brenti, independently) thatP(λ)2≥P(λ-1)P(λ+1)and also disproving several other conjectures of Brenti. Secondly, we prove that if the graph has n vertices, thenP(n)P(n-1)-1≥2.718253,approaching a conjecture of Bartels and Welsh thatP(n)P(n-1)-1≥e(eis 2.718281 ...).
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics