DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-07-31

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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

A bipartite graph and a proper two-colouring of its vertices

Definition

A graph $G=(V,E)$ is bipartite when there are disjoint subsets $A,B\subseteq V$ with $V=A\cup B$ such that every edge has one endpoint in $A$ and the other in $B$ . The ordered pair $(A,B)$ is a bipartition.

Equivalently, a proper two-colouring is a map $c:V\to\{0,1\}$ satisfying $c(u)\ne c(v)$ whenever $\{u,v\}\in E$ : take $A=c^{-1}[\{0\}]$ and $B=c^{-1}[\{1\}]$ , or define $c$ from a bipartition. Either part may be empty, so every edgeless graph, including the null graph, is bipartite.

Depends on

A finite simple graph is a finite vertex set together with a set of two-element vertex subsets

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 7 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

R. Diestel, Graph Theory, Chapter 1 preview (standard reference, not scraped)