Embeddings of four-valent framed graphs into two-surfaces

Framed $4$-valent Graph Minor Theory II: Special Minors and New Examples

In the present paper, we proceed the study of framed $4$-graph minor theory initiated in ``Framed $4$-valent Graph Minor Theory I. Intoduction. Planarity Criterion '' and justify the planarity theorem for arbitrary framed 4-graphs; besides, we prove analogous results for embeddability in the projective plane.

Framed 4-Valent Graphs: Euler Tours, Gauss Circuits and Rotating Circuits

In the present paper we give an explicit formula which allows us immediately to describe a unique Gauss circuit on a framed 4-valent graph (a graph with a structure of opposite edges) from an arbitrary Euler tour on the graph whenever the Gauss circuit exists. This formula only depends on the adjacency matrix of an Euler tour and also tells us whether there exists a Gauss tour on a framed 4-valent graph or not. It turns out that the results are also valid for all symmetric matrices (not just realisable by a chord diagram).

Framed 4-valent Graph Minor Theory I: Intoduction. A Planarity Criterion and Linkless Embeddability

The present paper is the first one in the sequence of papers about a simple class of {\em framed $4$-graphs}; the goal of the present paper is to collect some well-known results on planarity and to reformulate them in the language of {\em minors}. The goal of the whole sequence is to prove analogues of the Robertson-Seymour-Thomas theorems for framed $4$-graphs: namely, we shall prove that many minor-closed properties are classified by finitely many excluded graphs. From many points of view, framed $4$-graphs are easier to consider than general graphs; on the other hand, framed $4$-graphs are …

Embeddings of *-graphs into 2-surfaces

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A *-graph is a graph endowed with a formal adjacency structure on the half-edges around each vertex, and an embedding of a *-graph is an embedding under which the formal adjacency relation on half-edges corresponds to the adjacency relation induced by the embedding. *-graphs are a natural generalization of four-valent framed graphs, which are four-valent graphs with an opposite half-edge structure. In [5], the…

A new proof of Vassiliev's conjecture

We give a new proof of Vassiliev's planarity criterion for framed four-valent graphs (and more generally, *-graphs), which is based on Pontryagin-Kuratowski theorem.

映射空间的图示组合：配置空间类型的模之间的映射空间

Graphical composition of mapping spaces between modules of configuration-space-type

本文提出了一种组合图示方法，用于描述配置空间类型模之间的映射空间的组合结构，并探讨了余诱导函子在相关余模上的作用。