Unimodality of the odd-indexed Euler-genus distribution

Establish that, for every signed graph \(\Sigma\), the odd-indexed subsequence \(\left(g_{2j+1}(\Sigma)\right)_{j\geq 0}\) of its Euler-genus distribution is unimodal.

Background

The Euler-genus distribution of a signed graph counts embeddings by Euler genus. Although the paper characterizes the possible Euler genera and proves that each parity class is gap-free, it leaves unresolved the shape of the counts within each parity class.

The odd-indexed conjecture is the parity-counterpart to the proposed even-indexed conjecture. The paper states that the two conjectures are equivalent whenever both parity subsequences are nonempty.

References

\begin{conjecture} For every signed graph $\Sigma$, the odd-indexed subsequence $ \left(g_{2j+1}(\Sigma)\right)_ {j\geq 0} $ of its Euler-genus distribution is unimodal. \end{conjecture}

— Generalized Duke's theorem for signed Graphs  (2609.27628 - Chen et al., 23 Sep 2026) in Section 5, Conclusions