Final answer:
An Eulerian circuit is a path that visits every edge exactly once and starts and ends at the same vertex. For a graph K to have an Eulerian circuit, all the vertices must have even degrees.
Step-by-step explanation:
An Eulerian circuit is a path in a graph that visits every edge exactly once and starts and ends at the same vertex. For a graph K to have an Eulerian circuit, all the vertices in K must have even degrees.
An Eulerian trail is a path in a graph that visits every edge exactly once but starts and ends at different vertices. For a graph K to have an Eulerian trail, exactly two vertices in K must have odd degrees.
A Hamiltonian cycle is a cycle in a graph that visits every vertex exactly once. A Hamiltonian cycle exists in a graph if and only if every pair of non-adjacent vertices in K has a degree of at least 2.
A Hamiltonian path is a path in a graph that visits every vertex exactly once. A Hamiltonian path exists in a graph if and only if every pair of non-adjacent vertices in K has a degree of at least 1.