The minor minimal intrinsically chiral graphs

On intrinsically knotted or completely 3-linked graphs

We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link any of whose 2-component sublink is nonsplittable. We show that a graph obtained from the complete graph on seven vertices by a finite sequence of $\triangle Y$-exchanges and $Y \triangle$-exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph.

Bipartite intrinsically knotted graphs with 23 edges

A graph is intrinsically knotted if every embedding contains a nontrivially knotted cycle. It is known that intrinsically knotted graphs have at least 21 edges and that there are exactly 14 intrinsically knotted graphs with 21 edges, in which the Heawood graph is the only bipartite graph. The authors showed that there are exactly two graphs with at most 22 edges that are minor minimal bipartite intrinsically knotted: the Heawood graph and Cousin 110 of the $E_9+e$ family. In this paper we show that there are exactly six bipartite intrinsically knotted graphs with 23 edges so that every vertex …

Intrinsically projectively linked graphs

A graph is intrinsically projectively linked (IPL) if its every embedding in projective space contains a nonsplit link. Some minor-minimal IPL graphs have been found previously. We determine that no minor-minimal IPL graphs on 16 edges exists and identify new minor-minimal IPL graphs by applying $Δ-Y$ exchanges to $K_{7}-2e$. We prove that for a nonouter-projective-planar graph $G$, $G+\bar{K}_{2}$ is IPL and describe the necessary and sufficient conditions on a projective planar graph $G$ such that $G+\bar{K}_{2}$ is IPL. Lastly, we deduce conditions for $f(G + \bar{K_{2}})$ to have no nonspl…

An Algorithm for Detecting Intrinsically Knotted Graphs

We describe an algorithm that recognizes some (perhaps all) intrinsically knotted (IK) graphs, and can help find knotless embeddings for graphs that are not IK. The algorithm, implemented as a Mathematica program, has already been used by Goldberg, Mattman, and Naimi [6] to greatly expand the list of known minor minimal IK graphs, and to find knotless embeddings for some graphs that had previously resisted attempts to classify them as IK or non-IK.

Chirality for simple graphs of size up to 12

Chirality is one of the important assymmetrical property in wide area of natural science, which has been studied to predict molecular behavior. One of good methods to analyze molecules with complex structures is representing them as graphs embedded in 3-dimensional space. So it is important to study the chirality of spatial graphs to understand structure of chiral molecules. Moreover, Robertson and Seymour's graph minor theorem implies that a set of minor minimal graphs with respect to intrinsic properties is finite. So it is also important to find a complete set of minor minimal graphs for in…

Intrinsically Knotted and 4-Linked Directed Graphs

We consider intrinsic linking and knotting in the context of directed graphs. We construct an example of a directed graph that contains a consistently oriented knotted cycle in every embedding. We also construct examples of intrinsically 3-linked and 4-linked directed graphs. We introduce two operations, consistent edge contraction and H-cyclic subcontraction, as special cases of minors for digraphs, and show that the property of having a linkless embedding is closed under these operations. We analyze the relationship between the number of distinct knots and links in an undirected graph $G$ an…

非分离平面图补图中的完全子式

Complete Minors in Complements of Non-Separating Planar Graphs

本文证明阶为$2n-3$的非分离平面图补图必含$K_n$子式且该阶界最优，并通过反例、内在纽结论证及轮图补图Hadwiger数计算完善结论。

Intrinsic Chirality of Graphs in 3-manifolds

The main result of this paper is that for every closed, connected, orientable, irreducible 3-manifold $M$, there is an integer $ n_M$ such that any abstract graph with no automorphism of order 2 which has a 3-connected minor whose genus is more than $n_M$ has no achiral embedding in $M$. By contrast, the paper also proves that for every graph $γ$, there are infinitely many closed, connected, orientable, irreducible 3-manifolds $M$ such that some embedding of $γ$ in $M$ is pointwise fixed by an orientation reversing involution of $M$.

Graphs on 21 edges that are not 2--apex

We show that the 20 graph Heawood family, obtained by a combination of triangle-Y and Y-triangle moves on $K_7$, is precisely the set of graphs of at most 21 edges that are minor minimal for the property not $2$--apex. As a corollary, this gives a new proof that the 14 graphs obtained by triangle-Y moves on $K_7$ are the minor minimal intrinsically knotted graphs of 21 or fewer edges. Similarly, we argue that the seven graph Petersen family, obtained from $K_6$, is the set of graphs of at most 17 edges that are minor minimal for the property not apex.

On chirality of toroidal embeddings of polyhedral graphs

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1],[2]. Building on this and using the chirality of torus knots and links [3],[4], we prove that nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph emb…

射影平面图中的链环

Links in projective planar graphs

本文研究了射影平面图中的链环问题，部分刻画了子式极小分离射影平面图，并推广至I型3-链环，部分刻画了子式极小内蕴射影平面I型3-链环（IPPI3L）图。

Heawood 族的拓扑对称性

Topological Symmetries of the Heawood family

本文研究 Heawood 族图在 S³ 中的拓扑对称群，完整确定各图可实现的对称群，并据此证明该族所有图均为本征手性。

嵌入平面图中的良 quasi-序

Well-quasi-orders on embedded planar graphs

本文研究了嵌入平面图中经典minor关系的变种，证明其在不同约束条件下均为良quasi-序，拓展了拓扑图论的理论基础。