science

Wiener–Araya graph

Wiener–Araya graph is a science 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 Wiener–Araya graph rather than just read about it. In short: The Wiener–Araya graph is, in graph theory, a graph on 42 vertices with 67 edges. It is hypohamiltonian, which means that it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian.

Key takeaways

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

Reference excerpt

The Wiener–Araya graph is, in graph theory, a graph on 42 vertices with 67 edges. It is hypohamiltonian, which means that it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It is also planar.

History Hypohamiltonian graphs were first studied by Sousselier in Problèmes plaisants et délectables (1963). In 1967, Lindgren built an infinite sequence of hypohamiltonian graphs. He first cited Gaudin, Herz and Rossi, then Busacker and Saaty as pioneers on this topic. From the start, the smallest hypohamiltonian graph is known: the Petersen graph. However, the hunt for the smallest planar hypohamiltonian graph continues. This question was first raised by Václav Chvátal in 1973. The first candidate answer was provided in 1976 by Carsten Thomassen, who exhibited a 105-vertices construction, the 105-Thomassen graph. In 1979, Hatzel improved this result with a planar hypohamiltonian graph on 57 vertices : the Hatzel graph. This bound was lowered in 2007 by the 48-Zamfirescu graph. In 2009, a graph built by Gábor Wiener and Makoto Araya became (with its 42 vertices) the smallest planar hypohamiltonian graph known. In their paper, Wiener and Araya conjectured their graph to be optimal arguing that its order (42) appears to be the answer to The Ultimate Question of Life, the Universe, and Everything from The Hitchhiker's Guide to the Galaxy, a Douglas Adams novel. However, subsequently, smaller planar hypohamiltonian graphs have been discovered.

References

External links Weisstein, Eric W., "Wiener-Araya Graph", MathWorld

Illustrations

Worked examples

Example 1 — a first encounter with Wiener–Araya graph

Start with the simplest possible case. Write down what Wiener–Araya graph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Wiener–Araya 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 Wiener–Araya 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 Wiener–Araya graph

In research: Wiener–Araya graph appears in science 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 Wiener–Araya 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: Wiener–Araya graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Individual graphs, Planar graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Wiener–Araya 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.

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How to study Wiener–Araya graph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Wiener–Araya 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 Wiener–Araya graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Wiener–Araya graph in simple terms?

Why does Wiener–Araya graph matter?

Because it connects several science 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 Wiener–Araya 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 Wiener–Araya graph.