A proof using B\"ohme's Lemma that no Petersen family graph has a flat embedding

TL;DR

This paper provides an alternative proof, using Böhme's Lemma and the Jordan-Brouwer Separation Theorem, to show that no Petersen family graph admits a flat embedding, reinforcing their intrinsic linking properties.

Contribution

The paper introduces a novel proof technique for non-flatness of Petersen family graphs, differing from previous linking number and counting arguments.

Findings

All Petersen family graphs are intrinsically linked.

No Petersen family graph admits a flat embedding.

The proof leverages Böhme's Lemma and the Jordan-Brouwer Separation Theorem.

Abstract

Sachs and Conway-Gordon used linking number and a beautiful counting argument to prove that every graph in the Petersen family is intrinsically linked (have a pair of disjoint cycles that form a nonsplit link in every spatial embedding) and thus each family member has no flat spatial embedding (an embedding for which every cycle bounds a disk with interior disjoint from the graph). We give an alternate proof that every Petersen family graph has no flat embedding by applying B\"{o}hme's Lemma and the Jordan-Brouwer Separation Theorem.

Full text

1 Introduction

Sachs [5], [6] and Conway-Gordon [3] (for $K_{6}$) proved, using the linking number and a beautiful counting argument, that every graph in the Petersen family is intrinsically linked (that is, has a pair of disjoint cycles that form a nonsplit link in every spatial embedding). As a consequence of being intrinsically linked, every graph in the Petersen family has no flat embedding. A flat spatial embedding is one in which every cycle bounds a disk with interior disjoint from the graph. When such a disk is attached to a cycle, we say that cycle has been paneled.

We have devised a new family of proofs that every Petersen family graph has no flat embedding, without using intrinsic linking. We sketch our methods here. Given a graph $G$ with a flat embedding, consider a set of cycles in $G$ such that, pair-wise, intersections are either connected or empty. We call such a set of cycles a Böhme system. Böhme’s Lemma [2] concludes that for a Böhme system of cycles in a flat embedding of $G$, it is possible to panel the cycles in the system simultaneously–that is, each cycle bounds a disk with interior disjoint from the graph and from all other disks. The Jordan-Brouwer Separation Theorem then allows us to conclude that any sphere resulting from paneling a Böhme system (which we will call a Böhme sphere) will separate space into an inside (bounded) and an outside (unbounded) component. Since we work in the PL category, by Alexander’s result [1], the inside component is a topological ball.

For the sake of contradiction, we suppose a Petersen family graph has a flat embedding and then find an appropriate Böhme system. We then argue that some resulting Böhme sphere must be intersected by an edge of the embedded graph, which contradicts flatness. In this way, we prove that each Petersen family graph cannot have a flat embedding. In the next section, we provide further details for each graph in the Petersen Family.

We conclude this introductory section with some terminology and background. We work in the PL category throughout.

Suppose a graph $G$ has a three cycle with vertices $a,b,c$. To perform a $\Delta$ -Y exchange on the three-cycle in $G$, an extra vertex $y$ is added, edges $ab,bc,$ and $ca$ are removed, and the edges $ay,by,$ and $cy$ are added to transform $G$ into $G^{\prime}$. If a graph $H$ has a degree 3 vertex, $y$, to perform a Y-$\Delta$ exchange on $H$ at $y$, we begin with the Y, having vertices $a,b,c,y$ and edges $ay,by,$ and $cy$. We remove the $y$ vertex and all edges incident to $y$ and then add edges $ab,bc,$ and $ca$ to make a triangle and transform $H$ to the graph $H^{\prime}$.

The Petersen family of graphs consists of $K_{6}$ and $K_{3,3,1}$ and the graphs obtained from $K_{6}$ and $K_{3,3,1}$ by $\Delta-Y$ exchanges (equivalently, from $K_{6}$ by $\Delta-Y$ and $Y-\Delta$ exchanges). We will denote the remaining graphs in this family, with subscripts indicating the number of vertices, as $P_{7}$, $P_{8}$, $P_{9}$, $P_{10}$ (which is the classic Petersen graph) and $K_{4,4}-e$. Robertson, Seymour and Thomas proved Sachs’ conjecture that the Petersen family of graphs form the complete minor-minimal set of graphs that are intrinsically linked. They further showed that a graph has a flat embedding if and only if it has a linkless embedding [4].

2 Proofs

In this section, we go through details of the proof for each graph in the Petersen family.

2.1 $K_{6}$

Denote the vertices of $K_{6}$ as $\{a,b,c,d,e,f\}$. By way of contradiction suppose that $G=K_{6}$ has a flat embedding. Consider the collections of cycles:

$C_{1}=\{(b,e,f),(b,c,e),(c,e,f),(b,c,f)\}$

$C_{2}=\{(b,e,f),(a,b,e),(a,e,f),(a,b,f)\}$

$C_{3}=\{(b,e,f),(b,d,e),(d,e,f),(b,d,f)\}$

Let $\mathcal{C}=\cup^{3}_{i=1}C_{i}$. Note that the cycles above are induced cycles where $C_{1}\subset G[\{b,c,e,f\}],C_{2}\subset G[\{a,b,e,f\}]$, and $C_{3}\subset G[\{b,d,e,f\}]$, where $G[\{a,b,c,d\}]$ is the subgraph of $K_{6}$ induced by the vertices $a,b,c,d$. Notice $(b,e,f)\in C_{i}$ for $i=1,2,3$, see Figure 1.

Take a flat embedding of $K_{6}$. As all the cycles of $\mathcal{C}$ have connected pairwise intersections, we can apply Böhme’s Lemma to get simultaneous paneling disks of the ten 3-cycles that make up $\mathcal{C}$. As we panel each $C_{i}$, we form a sphere $T_{i}$, then by Jordan-Brouwer Theorem applied to, say, $T_{1}$, there are inside and outside components, $B_{1}$ and $B_{2}$ respectively, with $B_{1}\cup B_{2}={\mathbb{R}}^{3}-T_{1}$, and, say, $B_{1}$ a ball (which follows from Alexander’s result [1] in the PL category). Without loss of generality as the edge $\overline{ad}$ is disjoint from $T_{1}$, we may assume that $ad\subset B_{1}$. As $ad\subset B_{1}$ and $a\in T_{2}$, $d\in T_{3}$ then $T_{2},T_{3}\subset B_{1}\cup T_{1}$. This means that $T_{2}$ and $T_{3}$ are in the closure of $B_{1}$ (this is what we mean when we later say that $T_{2}$ and $T_{3}$ lie inside of $T_{1}$). Now similarly observe there exists, via Jordan-Brouwer, an inside and outside region for $T_{2}$. Without loss of generality suppose that $T_{3}$ is in the closure of the inner region (ball) of $T_{2}$. As $d\not\in T_{2}$ and $d\in T_{3}$, $d$ is in the inner ball of $T_{2}$. As $T_{2}$ is inside $T_{1}$ and $c\in T_{1}$, c is outside $T_{2}$. As edge $cd$ exists and as c and d are on opposite sides of $T_{2}$, edge $cd$ must intersect a disk of $T_{2}$. This brings us a contradiction. Thus $K_{6}$ has no flat embedding.

**The remaining cases

**For all remaining Petersen family graphs except $P_{10}$ and $K_{4,4}-e$, similarly to the proof for $K_{6}$, we consider a flat embedding, and a cycle $C$ with vertex set $V(C)$, together with three vertices disjoint from $V(C)$; $v_{1}$, $v_{2}$ and $v_{3}$, such that each $G[V(C)\cup\{v_{i}\}]$ forms a Böhme system. After panelling the Böhme system, the result is three spheres that intersect only along $C$ and the disk that panels $C$. In each case, two vertices are on opposite sides of a resulting Böhme sphere. In each case, the two vertices are connected either by an edge, or through a $Y$ that is disjoint from $G[V(C)\cup\{v_{i}\}]$ for the appropriate $i$, and the result is a contradiction of flatness. More details follow.

2.2 $P_{7}$

Denote the six vertices of $K_{6}$ by $\{a,b,c,1,2,3\}$, and suppose that a $\Delta-Y$ exchange took place between $1,2,$ and $3$. By way of contradiction, suppose that $P_{7}$ has a flat embedding. Consider the induced subgraphs:

$G_{1}=G[\{a,b,c,1\}]$

$G_{2}=G[\{a,b,c,2\}]$

$G_{3}=G[\{a,b,c,3\}]$

Note that each $G_{i}$ contains the cycle $(a,b,c)$. One can check that the induced cycles in $G_{1}\cup G_{2}\cup G_{3}=K_{3,1,1,1}$ form a Böhme system. Take a flat embedding of $P_{7}$. The simultaneous paneling of the Böhme system turns each $G_{i}$ into a Böhme sphere. From here, the argument follows analogously to the argument for $K_{6}$, except the vertices $\{1,2,3\}$ are connected by a $Y$ instead of a triangle.

2.3 $K_{4,4}-e$

Denote the vertices of $K_{4,4}-e$ as $V_{1}=\{a,b,c,d\}$ and $V_{2}\{1,2,3,4\}$ where all possible edges from $V_{1}$ to $V_{2}$ are present, except $e=4a$

Assume by way of contradiction that $K_{4,4}-e$ has a flat embedding.

Consider the set of cycles $A_{i}$ listed bellow, where $i\in\{1,2,3\}$.

$A_{1}=\{(1,b,4,c),(1,c,4,d),(1,b,4,d)\}\subseteq G[1,4,b,c,d]$

$A_{2}=\{(2,b,4,c),(2,c,4,d),(2,b,4,d)\}\subseteq G[2,4,b,c,d]$

$A_{3}=\{(3,b,4,c),(3,c,4,d),(3,b,4,d)\}\subseteq G[3,4,b,c,d]$

Notice that $A_{1}\cup A_{2}\cup A_{3}$ satisfies the hypotheses of Böhme’s Lemma with edges forming $K_{4,3}$. By Böhme’s lemma, we can simultaneously panel each of these cycles. Note that all paneled cycles pass through the vertex $4$. Denote the sphere resulting from paneling $A_{i}$ as $S_{i}$, where $i\in\{1,2,3\}$. Note that each $A_{i}$ contains the $Y$ on vertices $4,b,c,d$, as well as the vertex $i$. From here, the proof follows similarly to the argument used for $P_{7}$ ($K_{6}$), with the base cycle (triangle) $C$ replaced with the $Y$, where for $i\neq j$, $S_{i}\cap S_{j}$ is precisely the $Y$.

2.4 $K_{3,3,1}$

Label the vertices of $K_{3,3,1}$ as in Figure 4. Assume by way of contradiction that $K_{3,3,1}$ has a flat embedding. Consider the collection of cycles:

$\mathcal{S}_{1}=\{(a,2,c,3),(a,1,c,2),(a,1,c,3)\}\subseteq G[a,2,c,3,1]$

$\mathcal{S}_{2}=\{(a,2,c,3),(3,b,2,c),(3,b,2,a)\}\subseteq G[a,2,c,3,b]$

$\mathcal{S}_{3}=\{(a,2,c,3),(c,2,v),(c,v,3),(3,v,a),(a,v,2)\}\subseteq G[a,2,c,3,v]$

Notice that $\mathcal{S}_{1}\cup\mathcal{S}_{2}\cup\mathcal{S}_{3}$ satisfies Böhme’s lemma, and after paneling each, $\mathcal{S}_{i}$ forms a Böhme sphere. These Böhme spheres share precisely the same base cycle $(a,2,c,3)$ with the corresponding panelling disk. From here, the argument follows similarly to the argument for $K_{6}$.

2.5 $P_{8}$

Denote the vertices of $P_{8}$ by $\{v,a,b,c,1,2,3,y\}$, where $\{v\}$, $\{a,b,c\}$, and $\{1,2,3\}$ are the partitions in $K_{3,3,1}$, and triangle $(v,a,1)$ was exchanged to a $Y$, with resulting new $Y$ vertex $y$ in the resulting $P_{8}$.

By way of contradiction, suppose that $P_{8}$ is has a flat embedding. Consider the induced subgraphs:

$G_{1}=G[a,y,1,b,2,v]$

$G_{2}=G[a,y,1,b,2,c]$

$G_{3}=G[a,y,1,b,2,3]$

and their respective Böhme spheres formed by panelling their induced cycles (which is possible by Böhme’s Lemma):

$\mathcal{B}_{1}=\{(a,y,1,b,2),(v,2,a,y),(v,y,1,b),(v,b,2)\}$

$\mathcal{B}_{2}=\{(a,y,1,b,2),(a,y,1,c,2),(b,1,c,2)\}$

$\mathcal{B}_{3}=\{(a,y,1,b,2),(a,y,1,b,3),(a,2,b,3)\}$

Note that each $\mathcal{B}_{i}$ contains the cycle $(a,y,1,b,2)$. From here, the argument follows analogously to the argument for $K_{6}$.

2.6 $P_{9}$

Let $P_{9}$ be labelled as in Figure 6.

Assume by way of contradiction that $P_{9}$ is has a flat embedding. Consider a flat embedding, and consider the sets of cycles ($S_{i}$ where $i\in\{1,2,3\}$):

$S_{1}=\{(1,2,3,4,5,6),(6,7,3,4,5),(6,7,3,2,1)\}\subset G[1,2,3,4,5,6,7]$

$S_{2}=\{(1,2,3,4,5,6),(2,8,5,4,3),(2,8,5,6,1)\}\subset G[1,2,3,4,5,6,8]$

$S_{3}=\{(1,2,3,4,5,6),(1,9,4,3,2),(1,9,4,5,6)\}\subset G[1,2,3,4,5,6,9]$

Note that each $S_{i}$ contains the cycle $C=(1,2,3,4,5,6)$ and notice that by Böhme’s lemma each of these cycles can be simultaneously panelled so each $S_{i}$ determines a sphere for $i=1,2,3$ with, for $i\neq j$, the sphere determined by $S_{i}$ intersecting the sphere determined by $S_{j}$ only along $C$ and its corresponding panelling disk. From here, the proof follows analogously to the proof for $K_{6}$.

2.7 $P_{10}$

Suppose by way of contradiction that $P_{10}$ has a flat embedding. Consider the induced subgraphs formed by each set of vertices below, where labels are from Figure 7.

$G_{1}=G[\{1,2,3,4,5,6,9\}]$

$G_{2}=G[\{1,2,3,4,5,7,10\}]$

$G_{3}=G[\{1,2,3,4,5,6,8\}]$

Consider a flat embedding of $P_{10}$. Then by Böhme’s lemma, each 5-cycle and 6-cycle in $G_{1},G_{2},$ and $G_{3}$ can simultaneously bound disks to form three spheres, corresponding to each $G_{i}$. Note that the cycle $(1,2,3,4,5)$ is in each $G_{i}$.

By the Jordan-Brouwer Theorem, since vertex 7 is connected to vertex 10 by an edge, they must both be outside or both be inside of the sphere formed by $G_{1}$. Without loss of generality, assume both vertices are outside. Then the sphere created by $G_{2}$ is in the closure of the region outside of the sphere created by $G_{1}$. Since vertex 8 is connected to both 6 and 10 by an edge, vertex 8 must be between the sphere created by $G_{2}$ and the sphere created by $G_{1}$. Thus the sphere formed by $G_{3}$ is sandwiched between the sphere created by $G_{2}$ and the sphere created by $G_{1}$ (that is, it lies in the closure of the outer region formed by $G_{2}$ and in the closure of the inner region formed by $G_{1}$). But now since 7 is connected to 9 by an edge, and 7 is in $G_{2}$ and 9 is in $G_{1}$, the sphere formed by $G_{3}$ intersects the edge connecting 7 to 9. Hence a paneled disk must have been punctured by edge 79, contradicting flatness. In Figure 7, the panelled cycle $(2,3,4,10,8)$ from $G_{3}$ would be punctured. Therefore $P_{10}$ has no flat embedding.

We note that the $P_{10}$ case was similar to the earlier cases, except we needed a shared cycle $C$ and 5 additional vertices, and $G_{1}$ and $G_{3}$ shared the vertex $6$, in addition to the cycle $C$. The spheres they formed shared the base cycle (and disk) as well as the edge $5,6$. Instead of having a triangle or $Y$ connecting the extra vertices, here we have edges $8,10$ and $7,9$. Since $6$ lies in both $G_{1}$ and $G_{3}$ but not in $G_{2}$; $G_{1}$ and $G_{3}$ must be on the same side of $G_{2}$. Even with a swap of positions of $G_{1}$ and $G_{3}$ from our original proof, the edge $8,10$ would have intersected the Böhme sphere formed by $G_{1}$.

We could have done a $\Delta-Y$ exchange on $P_{9}$, on triangle $(7,8,9)$, to get $P_{10}$ with the same Böhme system we used on $P_{9}$ and the proof details analogous to those used on $P_{9}$.

3 Acknowledgements

This work was completed as part of a United States National Security Agency grant H89230-22-1-0008 Clarkson University: Summer (2022) Research Experience for Undergraduates in Mathematics.