mathematics

The Petersen Graph

The Petersen Graph is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand The Petersen Graph rather than just read about it. In short: The Petersen Graph is a mathematics book about the Petersen graph and its applications in graph theory. It was written by Derek Holton and John Sheehan, and published in 1993 by the Cambridge University Press as volume 7 in their Australian Mathematical Society Lecture Series.

Key takeaways

The Petersen Graph belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect The Petersen Graph to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of The Petersen Graph from memory before moving on to harder problems.

Reference excerpt

The Petersen Graph is a mathematics book about the Petersen graph and its applications in graph theory. It was written by Derek Holton and John Sheehan, and published in 1993 by the Cambridge University Press as volume 7 in their Australian Mathematical Society Lecture Series.

Topics

The Petersen graph is an undirected graph with ten vertices and fifteen edges, commonly drawn as a pentagram within a pentagon, with corresponding vertices attached to each other. It has many unusual mathematical properties, and has frequently been used as a counterexample to conjectures in graph theory. The book uses these properties as an excuse to cover several advanced topics in graph theory where this graph plays an important role. It is heavily illustrated, and includes both open problems on the topics it discusses and detailed references to the literature on these problems. After an introductory chapter, the second and third chapters concern graph coloring, the history of the four color theorem for planar graphs, its equivalence to 3-edge-coloring of planar cubic graphs, the snarks (cubic graphs that have no such colorings), and the conjecture of W. T. Tutte that every snark has the Petersen graph as a graph minor. Two more chapters concern closely related topics, perfect matchings (the sets of edges that can have a single color in a 3-edge-coloring) and nowhere-zero flows (the dual concept to planar graph coloring). The Petersen graph shows up again in another conjecture of Tutte, that when a bridgeless graph does not have the Petersen graph as a minor, it must have a nowhere-zero 4-flow. Chapter six of the book concerns cages, the smallest regular graphs with no cycles shorter than a given length. The Petersen graph is an example: it is the smallest 3-regular graph with no cycles of length shorter than 5. Chapter seven is on hypohamiltonian graphs, the graphs that do not have a Hamiltonian cycle through all vertices but that do have cycles through every set of all but one vertices; the Petersen graph is the smallest example. The next chapter concerns the symmetries of graphs, and types of graphs defined by their symmetries, including the distance-transitive graphs and strongly regular graphs (of which the Petersen graph is an example) and the Cayley graphs (of which it is not). The book concludes with a final chapter of miscellaneous topics too small for their own chapters.

Audience and reception The book assumes that its readers already have some familiarity with graph theory. It can be used as a reference work for researchers in this area, or as the basis of an advanced course in graph theory. Although Carsten Thomassen describes the book as "elegant", and Robin Wilson evaluates its exposition as "generally good", reviewer Charles H. C. Little takes the opposite view, finding fault with its copyediting, with some of its mathematical notation, and with its failure to discuss the lattice of integer combinations of perfect matchings, in which the number of copies of the Petersen graph in the "bricks" of a certain graph decomposition plays a key role in computing the dimension. Reviewer Ian Anderson notes the superficiality of some of its coverage, but concludes that the book "succeeds in giving an exciting and enthusiastic glimpse" of graph theory.

References

Illustrations

Worked examples

Example 1 — a first encounter with The Petersen Graph

Start with the simplest possible case. Write down what The Petersen Graph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to The Petersen Graph before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about The Petersen Graph ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of The Petersen Graph

In research: The Petersen Graph appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses The Petersen Graph in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: The Petersen Graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1993 non-fiction books, Mathematics books, so understanding it makes those chapters shorter.
In everyday life: Look for The Petersen Graph outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “The Petersen Graph” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study The Petersen Graph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what The Petersen Graph means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain The Petersen Graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is The Petersen Graph in simple terms?

Why does The Petersen Graph matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study The Petersen Graph?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on The Petersen Graph.