TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Every finite graph with $m$ edges has a cut containing at least $m / 2$ edges

Statement

Every finite simple graph with $m$ edges has a bipartition of its vertex set for which at least $m / 2$ edges have endpoints in different parts.

Facts & Assumptions

Given: A finite simple graph $G = (V, E)$ with $∣ E ∣ = m$ .

[L1]

A finite simple graph has a finite vertex set and two-element edges (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets).

[L2]

Independent fair coordinate choices form a finite product probability space (Product weights normalize, and coordinate events are mutually independent).

[L3]

Indicators turn an edge count into a sum, and expectation is linear without independence of those indicators (Indicators turn event probabilities, intersections, and finite counts into expectations and products, Expectation is linear for every finite family of random variables, without any independence hypothesis).

[L4]

Some outcome has value at least the expectation (Expectation preserves pointwise order and lies between the minimum and maximum attained values).

Proof

technique · direct

1.1

Place every vertex independently and fairly into one of two parts. A fixed edge crosses with probability $1 / 2$ .

L1L2

2.1

If $X$ is the number of crossing edges, [L3] gives $E [X] = m / 2$ .

step 1.1L3

3.1

By [L4], some bipartition has $X \geq m / 2$ . When $m = 0$ , every bipartition attains equality.

step 2.1L4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Matousek and J. Vondrak, The Probabilistic Method, Theorem 3.3.1 (standard reference, not scraped)
M. Bucic, Probabilistic Method, Theorem 2.1 (standard reference, not scraped)

Every finite graph with mm edges has a cut containing at least m/2m/2 edges