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Euler versus Hamilton circuit
Short answer
An Euler circuit traverses every edge exactly once and exists when the graph is connected and all vertices have even degree. A Hamiltonian circuit visits every vertex exactly once, but no simple necessary and sufficient condition exists; the problem is NP-complete, though sufficient conditions like Dirac's theorem (n≥3, min degree ≥ n/2) apply.
How does an Euler circuit (graph) differ from a Hamiltonian circuit (graph)? Explain.
An Euler circuit is a closed walk that uses every edge exactly once. A Hamiltonian circuit is a cycle that visits every vertex exactly once (and returns to the start).
| Point | Euler circuit | Hamiltonian circuit |
|---|---|---|
| Visits | every edge exactly once | every vertex exactly once |
| Vertices | may be repeated | no repetition (except start = end) |
| Edges | none repeated | some edges may be unused |
| Existence test | simple: connected and every degree even | no simple necessary and sufficient test (NP-complete) |
| Known conditions | exact (Euler's theorem) | only sufficient: Dirac (n ≥ 3, deg(v) ≥ n/2), Ore |
| Example | bowtie (two triangles sharing a vertex) has one, but no Hamiltonian circuit | K₄ has one, but no Euler circuit (all degrees 3) |
Bowtie: edges a–b, b–c, c–a, c–d, d–e, e–c. Euler circuit: c, a, b, c, d, e, c. No Hamiltonian circuit, since c would have to be visited twice.
Some graphs have both, for example the cycle C₅ or K₅.
✓ Answer checked by Claude Opus 5.5 (high effort)
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