Alphabeta MathHow 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.
satisfies but is nonplanar, so the planar edge bound is not sufficient
Statement refuted
Every finite simple graph with vertices and at most edges is planar.
Facts & Assumptions
Given: The complete bipartite graph of Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices.
Every simple planar graph with vertices has at most edges (Every simple planar graph with vertices has at most edges, with equality for every plane triangulation).
The graph is nonplanar ( and are nonplanar).
Counterexample
The two parts of contain three vertices each, so . Every vertex in either part is adjacent to all three vertices of the other part, giving ; equivalently this follows from Handshake lemma: the sum of the vertex degrees is twice the number of edges. Consequently , so the necessary inequality in [L1] holds.
Nevertheless [L2] says that this graph is nonplanar. Hence satisfying the planar simple-graph edge bound does not suffice for planarity.
Depends on
- Every simple planar graph with $n\ge3$ vertices has at most $3n-6$ edges, with equality for every plane triangulation
- $K_5$ and $K_{3,3}$ are nonplanar
- Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices
- Handshake lemma: the sum of the vertex degrees is twice the number of edges
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 46 results over 12 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, 6th ed., Corollary 4.2.11 (standard reference, not scraped)