Classification problem for real-rooted flow polynomials

The problem asks whether every bridgeless graph whose flow polynomial has only real zeros is the dual of a chordal plane graph. The claimed classification also restricts every zero to three integer values.

Known results

• Kung and Royle (2011): integral flow roots are equivalent to being the dual of a chordal plane graph.
• Dong (2018): the real-rooted classification is equivalent to excluding roots in $(1,2)$.
• Dong (2018): under connectivity hypotheses, a nonintegral real root forces at least $9$ roots in $(1,2)$.
• For $3$-connected cubic graphs, the real-root condition already implies the integral-root characterization.

August 2026 classification claim

Meiqiao Zhang and Fengming Dong claim that, for every bridgeless graph, real-rootedness is equivalent to integral roots and to being the dual of a chordal plane graph; they further claim that all roots lie in $\{1,2,3\}$. This would settle the classification, but the preprint has not been independently verified.

Current status (as of August 2026): A preprint claims the problem is solved, but its classification and proof remain unverified; no counterexample or independent confirmation was found.