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# Hamiltonian graph

A graph $G$ is *Hamiltonian* if it has a Hamiltonian cycle.

A useful condition both necessary and sufficient for a graph to be Hamiltonian is not known. Ore’s theorem and the Bondy and Chvátal theorem give sufficient conditions, while a necessary condition follows quickly from the definition, namely:

Let $G=(V,E)$ be a graph of order at least 3. If $G$ is Hamiltonian, then for every proper subset $U$ of $V$, the subgraph induced by $V-U$ has at most $|U|$ components.

Related:

HamiltonianCycle, HamiltonianPath, OresTheorem, BondyAndChvatalTheorem, PetersensGraph, Traceable

Synonym:

Hamiltonian

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

05C45*no label found*55-00

*no label found*82-00

*no label found*83-00

*no label found*81-00

*no label found*

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