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  "body": "\n## Definition\n\nLet $G$ be a finite [[graph|0JD5C82QMB]].\nA Hamiltonian path in $G$ is a [[path|0JDE56HWQ8]] in $G$ that contains every vertex of $G$.\n\n## Remarks\n\nSince a path visits each vertex at most once, a Hamiltonian path of a graph with $n$ vertices visits each vertex exactly once and has length $n - 1$.\nDeleting one edge from a [[Hamiltonian cycle|0HKXNG2RS1]] gives a Hamiltonian path, so every [[Hamiltonian graph|090WPKFABX]] has one.\nThe converse fails: the [[Petersen graph|0NR95CTFCZ]] has a Hamiltonian path but no Hamiltonian cycle.\nExample: in the [[cycle graph|0QQ0F12962]] $C_4$ with vertices $1, 2, 3, 4$ in cyclic order, the path $(1, 2, 3, 4)$ contains all four vertices, so it is a Hamiltonian path.\nNon-example: the star $K_{1,3}$ with centre $c$ and leaves $a, b, d$ has no Hamiltonian path.\nEach edge contains $c$, so a path alternates between $c$ and leaves, and since it visits $c$ at most once, it contains at most two leaves.\nA graph with a Hamiltonian path is connected.\n",
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