Half-integral flow polynomial values
Status open low confidence
Mohar's conjecture that $\Phi(G, 5.5) > 0$ for every 2-edge-connected graph $G$ appears to remain open. No post-2007 paper resolving the conjecture—either by proving it or finding a counterexample—was found across multiple literature searches. The problem asks for a non-integer rational $x > 5$ with this positivity property, going beyond Seymour's 6-flow theorem which only gives $\Phi(G,k) > 0$ for integer $k \geq 6$.
Reviewer notes. The canonical OPG page (both openproblemgarden.org and garden.irmacs.sfu.ca) was inaccessible (ECONNREFUSED). The survey arXiv:2007.05195 ('A survey on the study of real zeros of flow polynomials', 2020) is likely the most relevant post-2007 work but its full content was not retrievable to check for discussion of this specific conjecture. The ScienceDirect page for a note on flow polynomials (S0012365X08000721) returned HTTP 403. No paper was found that either proves or disproves the conjecture at x = 5.5. Confidence is low due to inability to access the full text of the most relevant survey.
Discussion
By Seymour's 6-flow theorem, $ \Phi(G,k) > 0 $ for every 2-edge-connected graph $ G $ and every integer $ k\ge6 $ . It would be interesting to find any non-integer rational number $ x>5 $ so that $ \Phi(G,x) > 0 $ for every 2-edge-connected graph $ G $ . It is known that zeros of flow polynomials are dense in the complex plane.