Gross–Mansour–Tucker's interpolating conjecture for partial-dual Euler-genus polynomials

Let $G$ be a non-orientable ribbon graph. Its partial-dual Euler-genus polynomial is the generating function

{}^{\partial}\varepsilon_G(z)=\sum_{A\subseteq E(G)}z^{\varepsilon(G^A)},

where $G^A$ is the partial dual of $G$ with respect to $A\subseteq E(G)$ and $\varepsilon$ denotes Euler genus. Gross–Mansour–Tucker's interpolating conjecture. The polynomial ${}^{\partial}\varepsilon_G(z)$ is interpolating. The conjecture was disproved by counterexamples, including two infinite classes of counterexamples found in this paper.

Sources & referencesView supporting material

Primary source

Qi Yan and Xian'an Jin, “Counterexamples to the interpolating conjecture on partial-dual genus polynomials of ribbon graphs”, arXiv:2009.05950 (2021).

1 week ago

Progress summary

Refreshed 6 days ago

Solved

The conjecture is false: explicit examples and two infinite families show that some non-orientable ribbon graphs do not have interpolating polynomials.

Gross, Mansour, and Tucker proved the orientable case and conjectured the analogous statement for every non-orientable ribbon graph. The conjecture is now disproved by explicit counterexamples.

Known results

Gross, Mansour, and Tucker: the orientable partial-dual genus polynomial is interpolating.
A smallest counterexample has four edges, with polynomial ${}^{\partial}\varepsilon_{\Theta}(z)=4z^{2}+12z^{4}$ .
Counterexamples were found among graphs with fewer than four edges for other related conjectures, not this one.

2020 counterexamples and 2022 extensions

The 2020 paper gives two infinite counterexample classes: $B_{2n+1}$ for every $n\ge 2$ and $C_{2n+2}$ for every $n\ge 1$ . A later account confirms these results and discusses additional bouquet families, including a non-interpolating class.

Current status (as of August 2026): The original conjecture is settled as false; related interpolation questions remain separate problems.

Sources

No solutions have been posted yet.