DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

How statement and proof provenance work

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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A decomposition of a graph's edge set into complete bipartite subgraphs

Definition

Let $G$ be a finite simple graph. A complete bipartite decomposition of $G$ is a list

$((X_{1}, Y_{1}), \dots, (X_{m}, Y_{m}))$

of pairs of disjoint nonempty subsets of $V (G)$ such that:

for each $k$ , every pair ${x, y}$ with $x \in X_{k}$ and $y \in Y_{k}$ is an edge of $G$ ; and
every edge of $G$ lies in exactly one of those complete bipartite edge sets.

So the decomposition is of the edge set of $G$ , not of its vertex set. The unions $X_{k} \cup Y_{k}$ need not cover all vertices.

Remarks

The phrase "exactly one" is the key point: the Graham-Pollak argument counts edges through those parts and needs a genuine partition of the edge set.

Depends on

Used by

Dependency tree · two levels

8 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

O. Pikhurko, An Introduction to Combinatorics, §3.2 (standard reference, not scraped)