Checkerboard colourable twisted duals

Excluded checkerboard colourable ribbon graph minors

In this paper, we first introduce the notions of checkerboard colourable minors for ribbon graphs motivated by the Eulerian ribbon graph minors, and two kinds of bipartite minors for ribbon graphs, one of which is the dual of the checkerboard colourable minors and the other is motivated by the bipartite minors of abstract graphs. Then we give an excluded minor characterization of the class of checkerboard colourable ribbon graphs, bipartite ribbon graphs, plane checkerboard colourable ribbon graphs and plane bipartite ribbon graphs.

Characterization of regular checkerboard colourable twisted duals of ribbon graphs

The geometric dual of a cellularly embedded graph is a fundamental concept in graph theory and also appears in many other branches of mathematics. The partial dual is an essential generalization which can be obtained by forming the geometric dual with respect to only a subset of edges of a cellularly embedded graph. The twisted dual is a further generalization by combining the partial Petrial. Given a ribbon graph $G$, in this paper, we first characterize regular partial duals of the ribbon graph $G$ by using spanning quasi-tree and its related shorter marking arrow sequence set. Then we chara…

On the ribbon graphs of links in real projective space

Every link diagram can be represented as a signed ribbon graph. However, different link diagrams can be represented by the same ribbon graphs. We determine how checkerboard colourable diagrams of links in real projective space, and virtual link diagrams, that are represented by the same ribbon graphs are related to each other. We also find moves that relate the diagrams of links in real projective space that give rise to (all-A) ribbon graphs with exactly one vertex.

嵌入图的Penrose多项式

A Penrose polynomial for embedded graphs

我们扩展彭罗斯多项式，原来只定义为平面图，图形嵌入在任意表面。考虑嵌入图的Penrose多项式，得到了嵌入图的Penrose多项式在平面图中不能实现的新的恒等式和关系式.特别是，通过利用与过渡多项式和带状群作用的连接，我们发现了一个删除收缩型的彭罗斯多项式的关系。我们与彭罗斯多项式的定向棋盘列图的电路分区多项式的中间图，并使用此找到新的组合解释的彭罗斯多项式。我们还证明了平面图G的Penrose多项式可以表示为G的扭曲图的色多项式之和。这使我们能够得到一个新的四色定理的重新表述。