部分对偶多项式:从带子图到delta-拟阵

Partial-dual polynomials:from ribbon graphs to delta-matroids

图的曲面嵌入是拓扑图论的一个主要研究内容,而几何对偶是拓扑图论中非常普遍且重要的概念.带子图是胞腔嵌入图的一种几何表示,可通过它引入部分对偶,即比几何对偶更加广泛的概念.部分对偶进一步被推广到部分twuality,它在多个领域均有广泛的应用,特别是拓扑和代数图论、拟阵论、拓扑学和物理学等.2020年,Gross等提出了带子图部分对偶欧拉亏格多项式,讨论了该多项式的基本性质并提出了若干问题和猜想.本文介绍本团队近几年在部分对偶欧拉亏格多项式方面的研究进展,包括推广这类多项式的概念和部分性质至delta-拟阵.

As one of primary objectives of topological graph theory,graph embeddings on surfaces are studied.Also,the geometric duality is regarded as a very pervasive and important concept in topological graph theory.Ribbon graphs are infact equivalent to cellularly embedded graphs.Also,in most studies on partial duality known as an extension of the geometric-duality concept,the language of ribbon graphs is often used.The concept of partial duality is further extended to the partial twuality,which arises in numerous areas,particularly topological and algebraic graph theory,matroid theory,topology,and physics.In 2020,Gross et al.introduced the partial-dual Euler-genus polynomial of a ribbon graph and discussed the fundamental properties of this new polynomial and proposed some problems and conjectures.In this paper,we report our recent progress on partial-dual Euler-genus polynomials and consider analogues of partial-dual Euler-genus polynomials for delta-matroids.