The Intrinsic Linking of Petersen Family Graphs

In mathematics, we often study graphs, which are structures made up of points (or vertices) connected by lines (or edges). Graphs can be represented in different ways, including in space, where they can be drawn without their edges crossing each other. A key aspect of this study is understanding whether certain types of graphs can be drawn flatly in space. In this article, we will explore why a special group of graphs, called the Petersen Family graphs, cannot be drawn flatly.

#Petersen Family Graphs

The Petersen Family consists of several graphs, including the well-known Petersen Graph. These graphs are notable because they have certain properties that make them interesting to mathematicians. One important property is that they are intrinsically linked, meaning there are pairs of cycles within these graphs that are linked in every possible spatial drawing. This property makes it impossible to separate these cycles without them intersecting.

#Flat Embeddings

A flat embedding of a graph means that every cycle within the graph can be enclosed in a disk without any intersections. In simpler terms, if you could stretch a piece of paper flat and draw the graph without any lines crossing, that would be a flat embedding. For a graph to have a flat embedding, it must be possible to draw it in such a way that each cycle (closed loop) can be surrounded by a flat circle, leaving no part of the graph inside the circle.

#Intrinsic Linking and Flatness

Because the graphs in the Petersen Family are intrinsically linked, they cannot be drawn flatly. If a graph has any linked cycles, it means that you cannot draw it without some cycles crossing over others, which prevents the possibility of a flat embedding. A significant result in graph theory states that if a graph is intrinsically linked, it cannot have a flat embedding.

#B ohme's Lemma

To prove that the Petersen Family graphs cannot be drawn flatly, we can use a concept called B ohme's Lemma. This lemma deals with collections of cycles within a graph. It states that if you have a set of cycles that either touch or do not touch each other, you can surround all of them with disks that do not intersect with the graph itself. This setup allows us to consider the arrangement of these cycles in space.

#Jordan-Brouwer Separation Theorem

Another useful concept in our proof is the Jordan-Brouwer Separation Theorem. This theorem tells us that when you draw a sphere around certain points in space, you create two distinct regions: an inside and an outside. Points on the inside cannot reach points on the outside without crossing the sphere's boundary. This principle helps us understand the relationships between the cycles in the graphs we are studying.

#Proving Non-Flatness

To prove that the Petersen Family graphs cannot be drawn flatly, we start by assuming that one of them can be. We then identify sets of cycles that must behave according to the B ohme's Lemma. We can then create spheres around these cycles and analyze their spatial relationships.

If we assume that a Petersen Family graph has a flat drawing, we can select cycles that intersect in certain ways. Using B ohme's Lemma, we try to surround these cycles with disks. However, due to the properties of the Petersen Family graphs, we will eventually reach a contradiction. This means that our initial assumption-that the graph can be drawn flatly-must be false.

#Connectedness of Cycles

In our proof, we examine how the cycles within the Petersen Family graphs are connected. If two cycles share an edge or are linked in any way, we cannot separate them purely based on the assumption of flatness. Each cycle must be fully contained within either the inside or outside of the sphere we create around them, according to the Jordan-Brouwer theorem. This interconnectedness leads to contradictions when we assume a flat embedding exists.

#Example of Contradiction

Let’s consider a scenario with a specific graph from the Petersen Family. We assume this graph can be drawn flatly. We then identify several cycles and assess how they interact with one another. By applying our earlier principles, we find that while some cycles are supposed to be enclosed in separate spheres, their connections force them into the same region. This results in an edge crossing from inside one sphere to outside another, which contradicts the definition of flatness.

#Generalizing the Argument

This reasoning can be generalized across all graphs in the Petersen Family. By systematically analyzing the relationships between the cycles and applying the principles of B ohme's Lemma and the Jordan-Brouwer theorem, we can consistently arrive at contradictions. Each graph will reveal its inability to maintain a flat embedding due to its intrinsic linking property.

#Conclusion

In summary, the Petersen Family graphs present a fascinating case in the study of graph theory. Their intrinsic linking prevents them from being drawn flatly, as proven by applying B ohme's Lemma and the Jordan-Brouwer Separation Theorem. By examining the nature of cycles within these graphs, we find that any assumption of flatness leads to contradictions. Therefore, we conclude that these graphs cannot have flat embeddings, which highlights the complex relationships within these interesting mathematical structures.