science

Unfriendly partition

Unfriendly partition 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 Unfriendly partition rather than just read about it. In short: In the mathematics of infinite graphs, an unfriendly partition or majority coloring is a partition of the vertices of the graph into disjoint subsets, so that every vertex has at least as many neighbors in other sets as it has in its own set. It is a generalization of the concept of a maximum cut for finite graphs, which is automatically an unfriendly partition.

Key takeaways

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

Reference excerpt

In the mathematics of infinite graphs, an unfriendly partition or majority coloring is a partition of the vertices of the graph into disjoint subsets, so that every vertex has at least as many neighbors in other sets as it has in its own set. It is a generalization of the concept of a maximum cut for finite graphs, which is automatically an unfriendly partition. (If not, a vertex with more neighbors in its own set could be moved to the other set, increasing the number of cut edges.) The unfriendly partition conjecture is an unsolved problem asking whether every countable graph has an unfriendly partition into two subsets. Robert H. Cowan and William R. Emerson, in unpublished work, conjectured that every infinite graph has an unfriendly partition into two subsets. However, Saharon Shelah and Eric Charles Milner disproved the conjecture, showing that uncountable graphs might not have two-subset unfriendly partitions. Nevertheless, they showed that an unfriendly partition into three subsets always exists. Among countable graphs, the existence of a two-subset unfriendly partition is known for the following special cases:

Graphs that have finitely many vertices of infinite degree Graphs in which all vertices have infinite degree, by an argument using the back-and-forth method Graphs with no end Graphs without a subdivision of an infinite clique The case for arbitrary countable graphs remains open.

References

Illustrations

Worked examples

Example 1 — a first encounter with Unfriendly partition

Start with the simplest possible case. Write down what Unfriendly partition 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 Unfriendly partition 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 Unfriendly partition 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 Unfriendly partition

In research: Unfriendly partition 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 Unfriendly partition 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: Unfriendly partition is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph theory objects, Infinite graphs, Unsolved problems in graph theory, so understanding it makes those chapters shorter.
In everyday life: Look for Unfriendly partition 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 Unfriendly partition in 20 minutes

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

Frequently asked questions

What is Unfriendly partition in simple terms?

Why does Unfriendly partition 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 Unfriendly partition?

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 Unfriendly partition.