TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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