Alphabeta MathGraph Colouring
About this page
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.
✓ 3 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Proper vertex colourings and chromatic number
Lemma
The greedy colouring bound χ(G)≤Δ(G)+1 for every nonnull finite graph
Definition
Cliques, independent sets, clique number and independence number
Theorem
The bounds ω(G)≤χ(G) and |V(G)|≤χ(G)α(G)
Definition
Kempe chains as connected components induced by two colour classes
Lemma