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.
and are nonplanar
Statement
The complete graph and complete bipartite graph of Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices are nonplanar.
Facts & Assumptions
Given: The two finite simple graphs and ; their degrees may also be counted with Handshake lemma: the sum of the vertex degrees is twice the number of edges.
The complete graph on vertices has exactly edges (The complete graph on an -element vertex set has edges).
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).
Every triangle-free simple planar graph with vertices has at most edges (Every triangle-free simple planar graph with vertices has at most edges).
Proof
By [L1], has edges, but [L2] would permit at most in a planar graph. Hence is nonplanar.
The graph has six vertices and nine edges. It is triangle-free because a closed walk alternates between its two parts and therefore has even length. A planar embedding would contradict [L3], whose bound is .
Depends on
- Every simple planar graph with $n\ge3$ vertices has at most $3n-6$ edges, with equality for every plane triangulation
- Every triangle-free simple planar graph with $n\ge3$ vertices has at most $2n-4$ edges
- Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices
- The complete graph on an $n$-element vertex set has $\binom{n}{2}$ edges
- Handshake lemma: the sum of the vertex degrees is twice the number of edges
Used by
Dependency tree · two levels
18 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, 6th ed., Corollary 4.2.11 (standard reference, not scraped)
- R. Grassl and O. Levin, Exploring Combinatorial Mathematics, Activities 295-296 (standard reference, not scraped)