mathematics

Thue number

Thue number 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 Thue number rather than just read about it. In short: In the mathematical area of graph theory, the Thue number of a graph is a variation of the chromatic index, defined by Alon et al. (2002) and named after mathematician Axel Thue, who studied the squarefree words used to define this number.

Key takeaways

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

Reference excerpt

In the mathematical area of graph theory, the Thue number of a graph is a variation of the chromatic index, defined by Alon et al. (2002) and named after mathematician Axel Thue, who studied the squarefree words used to define this number. Alon et al. define a nonrepetitive coloring of a graph to be an assignment of colors to the edges of the graph, such that there does not exist any even-length simple path in the graph in which the colors of the edges in the first half of the path form the same sequence as the colors of the edges in the second half of the path. The Thue number of a graph is the minimum number of colors needed in any nonrepetitive coloring. Variations on this concept involving vertex colorings or more general walks on a graph have been studied by several authors.

Example Consider a pentagon, that is, a cycle C 5 {\displaystyle C_{5}} of five vertices. If its edges are colored with two colors, some two adjacent edges will have the same color x {\displaystyle x} ; the path formed by those two edges will have the repetitive color sequence x x {\displaystyle xx} . If its edges are colored with three colors, one of the three colors will be used only once; the path of four edges formed by the other two colors will either have two consecutive edges or will form the repetitive color sequence x y x y {\displaystyle xyxy} . However, all repetitions can be avoided by using four colors, with one color repeated on two non-adjacent edges. Therefore, the Thue number of C 5 {\displaystyle C_{5}} is four.

Results Alon et al. use the Lovász local lemma to prove that the Thue number of any graph is at most quadratic in its maximum degree; they provide an example showing that for some graphs this quadratic dependence is necessary. In addition they show that the Thue number of a path of four or more vertices is exactly three, that the Thue number of any cycle is at most four, and that the Thue number of the Petersen graph is exactly five. The cycles C 5 {\displaystyle C_{5}} , C 7 {\displaystyle C_{7}} , C 9 {\displaystyle C_{9}} , C 10 {\displaystyle C_{10}} , C 14 {\displaystyle C_{14}} , and C 17 {\displaystyle C_{17}} have Thue number four. Resolving a conjecture of Alon et al., Currie showed that all other cycles have Thue number three.

Computational complexity Testing whether a coloring has a repetitive path is in NP, so testing whether a coloring is nonrepetitive is in co-NP, and Manin showed that it is co-NP-complete. The problem of finding such a coloring belongs to Σ 2 P {\displaystyle \Sigma _{2}^{P}} in the polynomial hierarchy, and again Manin showed that it is complete for this level.

Notes

References Alon, Noga; Grytczuk, Jaroslaw; Hałuszczak, Mariusz; Riordan, Oliver (2002). "Nonrepetitive colorings of graphs" (PDF). Random Structures & Algorithms. 21 (3–4): 336–346. doi:10.1002/rsa.10057. MR 1945373. S2CID 5724512. Barát, János; Varjú, P. P. (2008). "On square-free edge colorings of graphs". Ars Combinatoria. 87: 377–383. MR 2414029. Barát, János; Wood, David (2005). "Notes on nonrepetitive graph colouring". Electronic Journal of Combinatorics. 15 (1). R99. arXiv:math.CO/0509608. Bibcode:2005math......9608B. MR 2426162. Brešar, Boštjan; Klavžar, Sandi (2004). "Square-free coloring of graphs". Ars Combin. 70: 3–13. MR 2023057. Currie, James D. (2002). "There are ternary circular square-free words of length n for n ≥ 18". Electronic Journal of Combinatorics. 9 (1). N10. doi:10.37236/1671. MR 1936865. Kündgen, André; Pelsmajer, Michael J. (2008). "Nonrepetitive colorings of graphs of bounded tree-width". Discrete Mathematics. 308 (19): 4473–4478. doi:10.1016/j.disc.2007.08.043. MR 2433774. Manin, Fedor (2007). "The complexity of nonrepetitive edge coloring of graphs". arXiv:0709.4497 [cs.CC]. Schaefer, Marcus; Umans, Christopher (2005). "Completeness in the polynomial-time hierarchy: a compendium".

Further reading Grytczuk, Jarosław (2007). "Nonrepetitive colorings of graphs—a survey". International Journal of Mathematics and Mathematical Sciences. 2007. Art. ID 74639. doi:10.1155/2007/74639. MR 2272338.

External links Media related to Thue number at Wikimedia Commons

Illustrations

Worked examples

Example 1 — a first encounter with Thue number

Start with the simplest possible case. Write down what Thue number 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 Thue number 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 Thue number 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 Thue number

In research: Thue number 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 Thue number 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: Thue number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combinatorics on words, Graph coloring, Graph invariants, so understanding it makes those chapters shorter.
In everyday life: Look for Thue number 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 Thue number in 20 minutes

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

Frequently asked questions

What is Thue number in simple terms?

Why does Thue number 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 Thue number?

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 Thue number.