As the title of the question states, a proof for this proposition will be highly appreciated. The proof can either be inductive or, explained in plain English.
2026-03-27 23:35:28.1774654528
For a graph G with greater than or equal to 3 vertices, prove that G is Hamitonian if there is a Hamiltonian path between every pair of vertices
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Consider two adjacent verices $A$ and $B$ and let $p$ be the path connecting them by hypothesis. Since it must begin in $A$ and end in $B$, $p$ does not contain the edge $e$ connecting $A$ and $B$. Hence, it is enough to consider $p\cup\{e\}$.