DefinitionDefinition: AI-adaptedProof: Not applicableaudited 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 $c (u) \neq c (v)$ 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 · two levels

2 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

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