CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passverified 2026-08-04 (gpt-5.6-sol-codex-subscription)

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A connected graph with pairwise distinct edge weights has a unique minimum spanning tree

Statement

If a connected graph has pairwise distinct real edge weights, then it has exactly one minimum spanning tree.

Facts & Assumptions

Given: A connected weighted graph $(G,w)$ whose edge weights are pairwise distinct.

[F1]

At least one MST exists (Real edge-weighted graphs, total tree weight and minimum spanning trees).

[L2]

Exchanging an outside edge for an edge on its fundamental cycle preserves the spanning-tree property (The spanning-tree exchange lemma).

[L3]

Adding an edge outside a spanning tree creates its unique fundamental cycle (Every edge outside a spanning tree determines a unique fundamental cycle).

[F2]

Every finite nonempty set of edge weights has a least member, by order duality from Every nonempty finite set of reals has a maximum and a minimum.

Proof

technique · contradiction

1.1

Suppose $T$ and $T'$ are distinct MSTs. Choose the least-weight edge $e$ in the symmetric difference $E(T)\mathbin\triangle E(T')$ , and assume after swapping names that $e\in E(T)\setminus E(T')$ .

assume-contraF2

1.2

By [L3], $T'+e$ has a fundamental cycle. Apart from $e$ , that cycle cannot consist entirely of edges of $T$ , since then it would be a cycle in $T$ . Hence it contains an edge $f\in E(T')\setminus E(T)$ .

chooseL3

2.1

The choice of $e$ and distinct weights gives $w(e)<w(f)$ .

step 1.1step 1.2

3.1

The exchange $T'-f+e$ is a spanning tree of weight $w(T')-w(f)+w(e)<w(T')$ , contradicting the minimality of $T'$ .

step 1.2step 2.1L2F1

4.1

Hence the MST is unique.

step 3.1F1discharge-contradiction∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 49 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

ISI Bangalore discrete mathematics notes, Minimal spanning trees (standard reference, not scraped)