For example, in a complete undirected graph there exists a Hamiltonian cycle, so can we say that this graph has a Hamiltonian path as well?
Or whenever we have a cycle we can't have a path?
Thanks
2026-03-28 07:58:47.1774684727
Can a graph have both a Hamiltonian cycle and path at the same time?
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Yes, your first argument is correct. Suppose $G$ is a simple undirected graph with vertex set $V(G) = \{v_1,...,v_n\}$ and WLOG, say $v_1v_2...v_nv_1$ is a Hamiltonian cycle. Then, clearly, $v_1v_2...v_n$ is a Hamiltonian path. So, whenever a graph has a Hamiltonian cycle, it has a Hamiltonian path. Converse may not be true however.