science

Erdős–Gyárfás conjecture

Erdős–Gyárfás conjecture 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 Erdős–Gyárfás conjecture rather than just read about it. In short: In graph theory, the unproven Erdős–Gyárfás conjecture, made in 1995 by mathematician Paul Erdős and his collaborator András Gyárfás, states that every graph with minimum degree 3 contains a simple cycle whose length is a power of two. Erdős offered a prize of $100 for proving the conjecture, or $50 for a counterexample; it is one of many conjectures of Erdős.

Key takeaways

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

Reference excerpt

In graph theory, the unproven Erdős–Gyárfás conjecture, made in 1995 by mathematician Paul Erdős and his collaborator András Gyárfás, states that every graph with minimum degree 3 contains a simple cycle whose length is a power of two. Erdős offered a prize of $100 for proving the conjecture, or $50 for a counterexample; it is one of many conjectures of Erdős. If the conjecture is false, a counterexample would take the form of a graph with minimum degree three having no power-of-two cycles. It is known through computer searches of Gordon Royle and Klas Markström that any counterexample must have at least 17 vertices, and any cubic counterexample must have at least 30 vertices. Markström's searches found four graphs on 24 vertices in which the only power-of-two cycles have 16 vertices. One of these four graphs is planar; however, the Erdős–Gyárfás conjecture is now known to be true for the special case of 3-connected cubic planar graphs. The conjecture remains open for bipartite cubic graphs. This case has been approached through several related computational and structural directions. In work presented at the 14th Workshop on Graph Theory in Szklarska Poręba, Poland, in 2011, Pouria Salehi Nowbandegani and Hossein Esfandiari proved by computer search that a cubic bipartite counterexample must have at least 30 vertices. In 2026, Julius Tranquilli improved this lower bound to 60 vertices, using a certified exhaustive computation to show that every simple cubic bipartite graph on at most 58 vertices contains a cycle of length 4, 8, or 16. Consequently, any cubic bipartite counterexample to the Erdős–Gyárfás conjecture must have at least 60 vertices. The computation was independently checked using two exact procedures with different 16-cycle detection methods and a static witness certificate. The 2013 result of Heckman and Krakovski settles a different but overlapping cubic restriction—3-connected cubic planar graphs—but does not settle the general cubic bipartite case. Weaker results relating the degree of a graph to unavoidable sets of cycle lengths are known: there is a set S {\displaystyle S} of lengths, with | S | = O ( n 0.99 ) {\displaystyle |S|=O(n^{0.99})} , such that every graph with average degree ten or more contains a cycle with its length in S {\displaystyle S} , and every graph whose average degree is exponential in the iterated logarithm of n {\displaystyle n} necessarily contains a cycle whose length is a power of two. The conjecture is also known to be true for planar claw-free graphs and for graphs that avoid large induced stars and satisfy additional constraints on their degrees.

References

External links Exoo, Geoffrey, Graphs Without Cycles of Specified Lengths West, Douglas B., Erdős Gyárfás Conjecture on 2-power Cycle Lengths, Open Problems - Graph Theory and Combinatorics

Illustrations

Worked examples

Example 1 — a first encounter with Erdős–Gyárfás conjecture

Start with the simplest possible case. Write down what Erdős–Gyárfás conjecture 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 Erdős–Gyárfás conjecture 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 Erdős–Gyárfás conjecture 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 Erdős–Gyárfás conjecture

In research: Erdős–Gyárfás conjecture 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 Erdős–Gyárfás conjecture 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: Erdős–Gyárfás conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Conjectures, Paul Erdős, Unsolved problems in graph theory, so understanding it makes those chapters shorter.
In everyday life: Look for Erdős–Gyárfás conjecture 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 Erdős–Gyárfás conjecture in 20 minutes

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

Frequently asked questions

What is Erdős–Gyárfás conjecture in simple terms?

Why does Erdős–Gyárfás conjecture 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 Erdős–Gyárfás conjecture?

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 Erdős–Gyárfás conjecture.