Alphabeta MathEulerian and Hamiltonian Graphs: Examples and Counterexamples
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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.
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✓ 9 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 9 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
2 · Logical flowchart
3 · Items
Example
The Konigsberg bridges multigraph has four odd-degree vertices and no Euler trail
Counterexample
Two triangles sharing one vertex form an Eulerian graph with no Hamilton cycle
Counterexample
K₄ is Hamiltonian but has no Euler circuit
Counterexample
P₃ has a Hamilton path but no Hamilton cycle
Counterexample
Two disjoint copies of Kₘ show that Dirac's bound cannot be lowered to n/2-1 for even n=2m≥4
Example
K_3,3 has a Hamilton cycle
Example
K₄ with one new vertex adjacent to two clique vertices satisfies Ore's condition but not Dirac's condition
Example
A transitive tournament has a directed Hamilton path and no directed cycle
Example