How does the concept of a vertex cover relate to the existence of a Hamiltonian cycle in a graph?

In graph theory, a vertex cover is a set of vertices that covers all edges in a graph. The concept of a vertex cover is related to the existence of a Hamiltonian cycle in a graph because if a graph has a Hamiltonian cycle, then its vertex cover must include at least two vertices from each edge in the cycle. This is because a Hamiltonian cycle visits each vertex exactly once, so the vertices in the cycle must be covered by the vertex cover. Conversely, if a graph has a vertex cover that includes at least two vertices from each edge, it may indicate the potential existence of a Hamiltonian cycle in the graph.