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.
is Hamiltonian but has no Euler circuit
Statement refuted
Every Hamiltonian finite simple graph has an Euler circuit.
Counterexample
The complete graph has a Hamilton cycle but no Euler circuit.
Facts & Assumptions
Given: The complete graph on vertices .
A complete graph contains every edge between distinct vertices (Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices).
A Hamilton cycle visits every vertex exactly once before returning to its start (Hamilton paths, Hamilton cycles, Hamiltonian graphs and Hamilton-connected graphs).
Degree is the number of neighbours (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).
A finite simple graph can be regarded as the multigraph with the same vertex and edge sets and the evident endpoint map (Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph).
A connected finite multigraph has an Euler circuit exactly when every degree is even (Euler's theorem and Hierholzer's construction: a connected finite undirected multigraph has an Euler circuit if and only if every degree is even).
Verification
By completeness, is a cycle through all four vertices, so is Hamiltonian.
Every vertex is adjacent to the other three, so every degree is three. Regard as the corresponding multigraph from [F4]. It is connected, but its degrees are odd; [L1] therefore rules out an Euler circuit.
Hence Hamiltonicity does not imply existence of an Euler circuit.
Depends on
- Hamilton paths, Hamilton cycles, Hamiltonian graphs and Hamilton-connected graphs
- Euler's theorem and Hierholzer's construction: a connected finite undirected multigraph has an Euler circuit if and only if every degree is even
- Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices
- Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree
- Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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Sources
- Applied Combinatorics, Eulerian and Hamiltonian Graphs (standard reference, not scraped)