Star coloring

Key takeaways

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

Reference excerpt

In the mathematical field of graph theory, a star coloring of a graph G is a (proper) vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by Grünbaum (1973). The star chromatic number ⁠ χ s ( G ) {\displaystyle \chi _{s}(G)} ⁠ of G is the fewest colors needed to star color G.

In special classes of graphs Grünbaum (1973) observed that the star chromatic number is bounded for planar graphs. More precisely, the star chromatic number of planar graphs is at most 20, and some planar graphs have star chromatic number at least 10. More generally, the star chromatic number is bounded on every proper minor closed class. This result has been generalized to all low-tree-depth colorings (standard coloring and star coloring being low-tree-depth colorings with respective parameter 1 and 2). For every graph of maximum degree ⁠ d , {\displaystyle d,} ⁠ the star chromatic number is ⁠ O ( d 3 / 2 ) . {\displaystyle O(d^{3/2}).} ⁠ There exist graphs for which this bound is close to tight: they have star chromatic number ⁠ Ω ( d 3 / 2 / log 1 / 2 ⁡ n ) . {\displaystyle \Omega (d^{3/2}/\log ^{1/2}n).} ⁠

Complexity It is NP-complete to determine whether χ s ( G ) ≤ 3 {\displaystyle \chi _{s}(G)\leq 3} , even when G is a graph that is both planar and bipartite. Finding an optimal star coloring is NP-hard even when G is a bipartite graph.

Related concepts Star coloring is the special case for q = 3 {\displaystyle q=3} of q {\displaystyle q} -centered coloring, colorings in which every connected subgraph either uses at least q {\displaystyle q} colors or has at least one color that is used for exactly one vertex. For such a coloring, a connected subgraph with only two colors must be a star, with the vertex of a unique color at its center. There can be no edges between the remaining vertices in the component, because they would form two-vertex connected subgraphs without a uniquely used color. Another generalization of star coloring is the closely related concept of acyclic coloring, where it is required that every cycle uses at least three colors, so the two-color induced subgraphs are forests. If we denote the acyclic chromatic number of a graph G by ⁠ χ a ( G ) {\displaystyle \chi _{a}(G)} ⁠, we have that ⁠ χ a ( G ) ≤ χ s ( G ) {\displaystyle \chi _{a}(G)\leq \chi _{s}(G)} ⁠, and in fact every star coloring of G is an acyclic coloring. In the other direction, ⁠ χ s ( G ) ≤ 2 χ a ( G ) 2 − χ a ( G ) , {\displaystyle \chi _{s}(G)\leq 2\chi _{a}(G)^{2}-\chi _{a}(G),} ⁠ so each of the two kinds of chromatic number is bounded if and only if the other one is.

Notes

References Albertson, Michael O.; Chappell, Glenn G.; Kierstead, Hal A.; Kündgen, André; Ramamurthi, Radhika (2004), "Coloring with no 2-Colored P4's", The Electronic Journal of Combinatorics, 11 (1), doi:10.37236/1779, MR 2056078. Coleman, Thomas F.; Moré, Jorge (1984), "Estimation of sparse Hessian matrices and graph coloring problems" (PDF), Mathematical Programming, 28 (3): 243–270, doi:10.1007/BF02612334, hdl:1813/6374, MR 0736293. Fertin, Guillaume; Raspaud, André; Reed, Bruce (2004), "Star coloring of graphs", Journal of Graph Theory, 47 (3): 163–182, doi:10.1002/jgt.20029, MR 2089462. Grünbaum, Branko (1973), "Acyclic colorings of planar graphs", Israel Journal of Mathematics, 14 (4): 390–408, doi:10.1007/BF02764716, MR 0317982. Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2003), "Colorings and homomorphisms of minor closed classes", Discrete & Computational Geometry: The Goodman-Pollack Festschrift, Algorithms & Combinatorics, vol. 25, Springer-Verlag, pp. 651–664, MR 2038495. Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2006), "Tree depth, subgraph coloring and homomorphism bounds", European Journal of Combinatorics, 27 (6): 1022–1041, doi:10.1016/j.ejc.2005.01.010, MR 2226435. Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28, Springer, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058

External links Star colorings and acyclic colorings (1973), present at the Research Experiences for Graduate Students (REGS) at the University of Illinois, 2008.

Illustrations

Worked examples

Example 1 — a first encounter with Star coloring

Start with the simplest possible case. Write down what Star coloring claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In astronomy, 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 Star coloring 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 Star coloring 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 Star coloring

In research: Star coloring appears in astronomy 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 Star coloring 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: Star coloring is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph coloring, NP-complete problems, so understanding it makes those chapters shorter.
In everyday life: Look for Star coloring 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.

How to study Star coloring in 20 minutes

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

Frequently asked questions

What is Star coloring in simple terms?

In the mathematical field of graph theory, a star coloring of a graph G is a (proper) vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components t…

Why does Star coloring matter?

Because it connects several astronomy 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 Star coloring?

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 Star coloring.