6 problems

Stahl's real-rootedness conjecture. For every graph $G$, the genus polynomial $\Gamma_G(x)$ has only real roots.

Log-Concavity Genus Distribution conjecture. For every graph $G$, the genus polynomial is log-concave.

For each graph family listed in Example, let its genus polynomial be the polynomial whose coefficients count embeddings by genus. Stahl's conjecture. The zeros of the genus polynom…

A ribbon graph is orientable if its underlying surface is orientable, and its partial-dual genus polynomial records the genera of all partial duals. A polynomial is non-constant if…

Let $G$ be a non-orientable ribbon graph, and define its partial-dual Euler-genus polynomial by … A polynomial is interpolating when its non-zero coefficients are all equal to $1$.…

Let $(G,v)$ be a rooted graph. For an almost complete rotation system at $v$, let the local genus polynomial be the genus polynomial of the embeddings consistent with that system.…