Unimodality of the even-indexed Euler-genus distribution

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

Background

For a signed graph Σ\Sigma, gk(Σ)g_k(\Sigma) denotes the number of embeddings with Euler-genus kk. The paper proves that the even- and odd-indexed subsequences are gap-free, but it does not determine whether their values rise and then fall in a unimodal pattern.

The authors propose unimodality separately for the even- and odd-indexed subsequences. They note that the two conjectures are equivalent when both subsequences are nonempty.

References

Therefore, we propose the following two conjectures. We also note that two conjectures are equivalent when both even-indexed subsequence and odd-indexed subsequence are nonempty, that is, if one holds then so does the other.

\begin{conjecture} For every signed graph $\Sigma$, the even-indexed subsequence $ \left(g_{2j}(\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