science

Uniquely colorable graph

Uniquely colorable 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 Uniquely colorable graph rather than just read about it. In short: In graph theory, a uniquely colorable graph is a k-chromatic graph that has only one possible (proper) k-coloring up to permutation of the colors. Equivalently, there is only one way to partition its vertices into k independent sets and there is no way to partition them into k − 1 independent sets.

Key takeaways

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

Reference excerpt

In graph theory, a uniquely colorable graph is a k-chromatic graph that has only one possible (proper) k-coloring up to permutation of the colors. Equivalently, there is only one way to partition its vertices into k independent sets and there is no way to partition them into k − 1 independent sets.

Examples A complete graph is uniquely colorable, because the only proper coloring is one that assigns each vertex a different color. Every k-tree is uniquely (k + 1)-colorable. The uniquely 4-colorable planar graphs are known to be exactly the Apollonian networks, that is, the planar 3-trees. Every connected bipartite graph is uniquely 2-colorable. Its 2-coloring can be obtained by choosing a starting vertex arbitrarily, coloring the vertices at even distance from the starting vertex with one color, and coloring the vertices at odd distance from the starting vertex with the other color.

Properties A uniquely k-colorable graph G with n vertices has at least m ≥ (k−1)n − k(k−1)/2 edges. Equality holds when G is a (k−1)-tree.

Related concepts

Minimal imperfection A minimal imperfect graph is a graph in which every subgraph is perfect. The deletion of any vertex from a minimal imperfect graph leaves a uniquely colorable subgraph.

Unique edge colorability

A uniquely edge-colorable graph is a k-edge-chromatic graph that has only one possible (proper) k-edge-coloring up to permutation of the colors. The only uniquely 2-edge-colorable graphs are the paths and the cycles. For any k, the stars K1,k are uniquely k-edge-colorable. Moreover, Wilson (1976) conjectured and Thomason (1978) proved that, when k ≥ 4, they are also the only members in this family. However, there exist uniquely 3-edge-colorable graphs that do not fit into this classification, such as the graph of the triangular pyramid. If a cubic graph is uniquely 3-edge-colorable, it must have exactly three Hamiltonian cycles, formed by the edges with two of its three colors, but some cubic graphs with only three Hamiltonian cycles are not uniquely 3-edge-colorable. Every simple planar cubic graph that is uniquely 3-edge-colorable contains a triangle, but W. T. Tutte (1976) observed that the generalized Petersen graph G(9,2) is non-planar, triangle-free, and uniquely 3-edge-colorable. For many years it was the only known such graph, and it had been conjectured to be the only such graph but now infinitely many triangle-free non-planar cubic uniquely 3-edge-colorable graphs are known.

Unique total colorability A uniquely total colorable graph is a k-total-chromatic graph that has only one possible (proper) k-total-coloring up to permutation of the colors. Empty graphs, paths, and cycles of length divisible by 3 are uniquely total colorable graphs. Mahmoodian & Shokrollahi (1995) conjectured that they are also the only members in this family. Some properties of a uniquely k-total-colorable graph G with n vertices:

χ″(G) = Δ(G) + 1 unless G = K2. Δ(G) ≤ 2 δ(G). Δ(G) ≤ n/2 + 1. Here χ″(G) is the total chromatic number; Δ(G) is the maximum degree; and δ(G) is the minimum degree.

Notes

References Akbari, S. (2003), "Two conjectures on uniquely totally colorable graphs", Discrete Mathematics, 266 (1–3): 41–45, doi:10.1016/S0012-365X(02)00797-5, MR 1991705. Akbari, S.; Behzad, M.; Hajiabolhassan, H.; Mahmoodian, E. S. (1997), "Uniquely total colorable graphs", Graphs and Combinatorics, 13 (4): 305–314, doi:10.1016/S0012-365X(02)00797-5, MR 1485924. belcastro, sarah-marie; Haas, Ruth (2015), "Triangle-free uniquely 3-edge colorable cubic graphs", Contributions to Discrete Mathematics, 10 (2): 39–44, arXiv:1508.06934, doi:10.11575/cdm.v10i2.62320, MR 3499076. Bollobás, Béla (1978), Extremal Graph Theory, LMS Monographs, vol. 11, Academic Press, MR 0506522. Fowler, Thomas (1998), Unique Coloring of Planar Graphs (PDF), Ph.D. thesis, Georgia Institute of Technology Mathematics Department. Hillar, Christopher J.; Windfeldt, Troels (2008), "Algebraic characterization of uniquely vertex colorable graphs", Journal of Combinatorial Theory, Series B, 98 (2): 400–414, arXiv:math/0606565, doi:10.1016/j.jctb.2007.08.004, MR 2389606, S2CID 108304. Mahmoodian, E. S. (1998), "Defining sets and uniqueness in graph colorings: a survey", Journal of Statistical Planning and Inference, 73 (1–2): 85–89, doi:10.1016/S0378-3758(98)00053-6, MR 1655213. Mahmoodian, E. S.; Shokrollahi, M. A. (1995), "Open problems at the combinatorics workshop of AIMC25 (Tehran, 1994)", in C. J., Colbourn; E. S., Mahmoodian (eds.), Combinatorics Advances, Mathematics and its applications, vol. 329, Dordrecht; Boston; London: Kluwer Academic Publishers, pp. 321–324. Schwenk, Allen J. (1989), "Enumeration of Hamiltonian cycles in certain generalized Petersen graphs", Journal of Combinatorial Theory, Series B, 47 (1): 53–59, doi:10.1016/0095-8956(89)90064-6, MR 1007713. Thomason, A. G. (1978), "Hamiltonian cycles and uniquely edge colourable graphs", Advances in Graph Theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977), Annals of Discrete Mathematics, vol. 3, pp. 259–268, MR 0499124. Thomason, Andrew (1982), "Cubic graphs with three Hamiltonian cycles are not always uniquely edge colorable", Journal of Graph Theory, 6 (2): 219–221, doi:10.1002/jgt.3190060218, MR 0655209. Truszczyński, M. (1984), "Some results on uniquely colourable graphs", in Hajnal, A.; Lovász, L.; Sós, V. T. (eds.), Finite and Infinite Sets. Vol. I, II. Proceedings of the sixth Hungarian combinatorial colloquium held in Eger, July 6–11, 1981, Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, pp. 733–748, MR 0818274. Tutte, William T. (1976), "Hamiltonian circuits", Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo I, Accad. Naz. Lincei, Rome, pp. 193–199. Atti dei Convegni Lincei, No. 17, MR 0480185. As cited by belcastro & Haas (2015). Xu, Shao Ji (1990), "The size of uniquely colorable graphs", Journal of Combinatorial Theory, Series B, 50 (2): 319–320, doi:10.1016/0095-8956(90)90086-F, MR 1081235. Wilson, R. J. (1976), "Problem 2", in Nash-Williams, C. St. J. A.; Sheehan, J. (eds.), Proc. British Comb. Conf. 1975, Winnipeg: Utilitas Math., p. 696. As cited by Thomason (1978).

External links Weisstein, Eric W., "Uniquely Colorable Graph", MathWorld

Worked examples

Example 1 — a first encounter with Uniquely colorable graph

Start with the simplest possible case. Write down what Uniquely colorable 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 Uniquely colorable 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 Uniquely colorable 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 Uniquely colorable graph

In research: Uniquely colorable 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 Uniquely colorable 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: Uniquely colorable graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph coloring, so understanding it makes those chapters shorter.
In everyday life: Look for Uniquely colorable 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.

Affiliate

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

How to study Uniquely colorable graph in 20 minutes

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

Frequently asked questions

What is Uniquely colorable graph in simple terms?

Why does Uniquely colorable 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 Uniquely colorable 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 Uniquely colorable graph.