To prove that every ( k )-regular graph on ( 2k + 1 ) vertices is Hamiltonian, we’ll use some concepts and arguments from graph theory. A ( k )-regular graph means that every vertex has exactly ( k ) edges.
Here's how we can approach the proof step-by-step:
- ( k )-regular Graph: A graph where every vertex has exactly ( k ) edges.
- Number of Vertices: ( 2k + 1 ) vertices.
- Hamiltonian Cycle: A cycle that visits each vertex exactly once and returns to the starting vertex.