Importance: Medium ✭✭
Recomm. for undergrads: no
Posted by: mdevos
on: March 7th, 2007
Conjecture   All real roots of nonzero flow polynomials are at most 4.

For every graph $ G $, let $ P_G $ be the chromatic polynomial of $ G $ and let $ Q_G $ be the flow polynomial of $ G $. If $ G $ is loopless, then $ P_G(k)>0 $ for all sufficiently large integers $ k $ (as $ P_G(k) $ = # of k-colorings of $ G $). It follows from Seymour's 6-flow theorem that if $ G $ has no bridge, then $ Q_G(k)>0 $ for all integers $ k>5 $ (as $ Q_G(k) $ = # of nowhere-zero flows in the group of integers modulo $ k $). It is natural to ask if all real roots of these polynomials are small. For the chromatic polynomial, $ P_G $, this is not the case. There exist graphs with chromatic number 3 for which $ P_G $ has arbitrarily large real roots. The above conjecture asserts that the flow polynomial exhibits the opposite behavior. One word of caution, it is known that the set of roots of flow polynomials is dense in the complex plane.


[S] P.D. Seymour, Nowhere-Zero 6-Flows, J. Combinatorial Theory Ser. B 30 (1981) 130-135. MathSciNet

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