On the maximum twist width of delta-matroids
Xian'an Jin, Zhuo Li, Qi Yan, Gang Zhang
TL;DR
The paper extends the known link between partial duals of ribbon graphs and twists of their associated delta-matroids to establish a universal, monotone-twist framework. It proves that the maximum twist width $\partial\omega_M(D)$ is always attained by twisting a single feasible set in any delta-matroid $D$, and that for ribbon graphs this yields $\partial\gamma_M(G)=\partial\omega_M(D(G))$ with realization by a spanning quasi-tree. Further, it shows there exists a sequence of twists, drawn from a feasible set, that increases monotonically to the maximum, and provides an algorithm to construct such sequences. These results affirmatively answer the corresponding problem for ribbon graphs and generalize the maximum-genus attainment to the delta-matroid setting, offering a practical method to obtain nondecreasing sequences that reach the maximum partial-dual genus. The work solidifies the delta-matroid perspective as a robust tool for studying partial duals and genus in embedded graphs.
Abstract
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.