An Eulerian path is a path that visits every edge of a graph exactly once. It is not necessary for the path to visit every vertex of the graph.

An Eulerian circuit is a circuit that visits every vertex of a graph exactly once. It is not necessary for the circuit to visit every edge of the graph.

A cycle is a graph that consists of a single circuit. Since every vertex in a cycle has degree 2, it is possible to find an Eulerian circuit in a cycle.

A path is a graph that consists of a single path. Since the endpoints of a path have degree 1, it is not possible to find an Eulerian circuit in a path. However, it is possible to find an Eulerian path in a path by starting at one endpoint and ending at the other endpoint.

A tree is a graph that is connected and has no cycles. Since every vertex in a tree has degree 1 or 2, it is not possible to find an Eulerian path or an Eulerian circuit in a tree.

A Hamiltonian path is a path that visits every vertex of a graph exactly once. It is not necessary for the path to visit every edge of the graph.

A Hamiltonian circuit is a circuit that visits every vertex of a graph exactly once. It is not necessary for the circuit to visit every edge of the graph.

A complete graph is a graph in which every vertex is connected to every other vertex. Since every vertex in a complete graph has degree n-1, where n is the number of vertices in the graph, it is possible to find a Hamiltonian circuit in a complete graph.

A path is a graph that consists of a single path. Since the endpoints of a path have degree 1, it is not possible to find a Hamiltonian circuit in a path. However, it is possible to find a Hamiltonian path in a path by starting at one endpoint and ending at the other endpoint.

A tree is a graph that is connected and has no cycles. Since every vertex in a tree has degree 1 or 2, it is not possible to find a Hamiltonian path or a Hamiltonian circuit in a tree.

Euler's formula states that for a connected graph with v vertices and e edges, the following equation holds: v + e = f, where f is the number of faces in the graph.

A path is a graph that consists of a single path. Since the endpoints of a path have degree 1, the number of vertices in a path is always odd.

A complete graph is a graph in which every vertex is connected to every other vertex. Since every vertex in a complete graph has degree n-1, where n is the number of vertices in the graph, the number of vertices in a complete graph is always even.

A cycle is a graph that consists of a single circuit. Since every edge in a cycle is incident to two vertices, the number of edges in a cycle is always even.

A complete graph is a graph in which every vertex is connected to every other vertex. Since every vertex in a complete graph has degree n-1, where n is the number of vertices in the graph, the number of edges in a complete graph is always n(n-1)/2, which is even.