On the maximum twist width of delta-matroids

For a ribbon graph $G$, let $γ(G)$ denote its Euler genus. Recently, Chen, Gross and Tucker [J. Algebraic Combin. 63 (2026) 13] derived a formula for the maximum partial-dual Euler-genus $\partialγ_M(G)$ of a ribbon graph $G$. Their key finding is that $\partialγ_M(G)$ can be achieved by a partial dual with respect to the edge set of a spanning quasi-tree. Moreover, they proposed the following problem: Given a ribbon graph $G$, is there a sequence of edges $e_1,e_2,\dots, e_k$ such that $γ(G^{{e_1, e_2,\dots, e_k}})=\partialγ_M(G)$ and such that the sequence $$γ(G), γ(G^{{e_1}}), \dots, γ(G^ {{e_1, e_2,\dots, e_k}})$$ rises monotonically (i.e., never decreasing) to $\partialγ_M(G)$? Delta-matroids are set systems that satisfy the symmetric exchange axiom and serve as a matroidal abstraction of ribbon graphs. In this paper, we first show that the maximum twist width of a set system can be attained by twisting one of its feasible sets, which extends the result of Chen, Gross and Tucker to set systems. Then we solve the delta-matroid version of their problem, thereby providing an affirmative answer to the original problem for ribbon graphs.

💡 Research Summary

The paper investigates the relationship between ribbon graphs and delta‑matroids through the notion of “maximum twist width,” which corresponds to the maximum Euler genus attainable by taking partial duals of a ribbon graph. For a ribbon graph (G) the Euler genus is denoted (\gamma(G)). Chen, Gross and Tucker (2026) showed that the maximum partial‑dual Euler genus (\partial\gamma_M(G)=\max_{A\subseteq E(G)}\gamma(G^A)) is achieved by a partial dual with respect to the edge set of a spanning quasi‑tree. The authors of the present work extend this result from ribbon graphs to arbitrary set systems (delta‑matroids) and answer the monotonicity question posed by Chen et al.

The authors first define, for any set system (D=(E,\mathcal{F})), the twist width (\omega(D)=r_{\max}(D)-r_{\min}(D)), where (r_{\max}) and (r_{\min}) are the sizes of a largest and a smallest feasible set, respectively. The maximum twist width is (\partial\omega_M(D)=\max_{A\subseteq E}\omega(D\Delta A)). Theorem 3.1 proves that there always exists a feasible set (F\in\mathcal{F}) such that (\omega(D\Delta F)=\partial\omega_M(D)). The proof picks a feasible set (F) minimizing (|A\Delta F|) for an arbitrary (A) and repeatedly uses the symmetric exchange axiom to replace elements of (A) by those of (F) without decreasing the width. Specializing to ribbon graphs, Corollary 3.2 recovers the Chen‑Gross‑Tucker result: there is a spanning quasi‑tree (A) with (\gamma(G^A)=\partial\gamma_M(G)).

The second major contribution addresses the monotonicity problem. Lemma 4.1 (a known result) guarantees that for any feasible set (F_0) there exist a minimal feasible set (F_{\min}) and a maximal feasible set (F_{\max}) with (F_{\min}\subseteq F_0\subseteq F_{\max}). Using this, Theorem 4.2 shows that one can choose a feasible set (F) attaining the maximum twist width such that there exists an ordering of its elements (e_1,\dots,e_k) with the property that the sequence \