science

Gyárfás–Sumner conjecture

Gyárfás–Sumner 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 Gyárfás–Sumner conjecture rather than just read about it. In short: In graph theory, the Gyárfás–Sumner conjecture says that for any fixed pair of a tree T {\displaystyle T} and a complete graph K t {\displaystyle K_{t}} , every graph that is both T {\displaystyle T} -free and K t {\displaystyle K_{t}} -free is also χ {\displaystyle \chi } -bounded; that is, every graph that contains neither T {\displaystyle T} nor K t {\displaystyle K_{t}} as an induced subgraph can be properly col…

Key takeaways

Gyárfás–Sumner 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 Gyárfás–Sumner conjecture to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Gyárfás–Sumner conjecture from memory before moving on to harder problems.

Reference excerpt

In graph theory, the Gyárfás–Sumner conjecture says that for any fixed pair of a tree T {\displaystyle T} and a complete graph K t {\displaystyle K_{t}} , every graph that is both T {\displaystyle T} -free and K t {\displaystyle K_{t}} -free is also χ {\displaystyle \chi } -bounded; that is, every graph that contains neither T {\displaystyle T} nor K t {\displaystyle K_{t}} as an induced subgraph can be properly colored using a constant number of colors. Equivalently, it says that every K t {\displaystyle K_{t}} -free graph G {\displaystyle G} contains any arbitrarily chosen tree T {\displaystyle T} (as an induced subgraph), as long as χ ( G ) {\displaystyle \chi (G)} is large enough. It is named after András Gyárfás and David Sumner, who formulated it independently in 1975 and 1981 respectively. It remains unproven. In this conjecture, it is not possible to replace T {\displaystyle T} by a graph with cycles. As Paul Erdős and András Hajnal have shown, there exist graphs with arbitrarily large chromatic number and, at the same time, arbitrarily large girth. Using these graphs, one can obtain graphs that avoid any fixed choice of a cyclic graph and clique (of more than two vertices) as induced subgraphs, and exceed any fixed bound on the chromatic number. The conjecture is known to be true for certain special choices of T {\displaystyle T} , including paths, stars, and trees of radius two. It is also known that, for any tree T {\displaystyle T} , the graphs that do not contain any subdivision of T {\displaystyle T} are χ {\displaystyle \chi } -bounded.

References

External links Graphs with a forbidden induced tree are chi-bounded, Open Problem Garden

Worked examples

Example 1 — a first encounter with Gyárfás–Sumner conjecture

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

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

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Gyárfás–Sumner conjecture” →

Affiliate

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

How to study Gyárfás–Sumner conjecture in 20 minutes

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

Frequently asked questions

What is Gyárfás–Sumner conjecture in simple terms?

Why does Gyárfás–Sumner 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 Gyárfás–Sumner 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 Gyárfás–Sumner conjecture.