For what values of n dies the complete graph K_n with n vertices have (a) an Euler circuit? (b) a Hamiltonian circuit? Justify your answers.

(a)

For what values of n does the complete graph Kn with n vertices have an Euler circuit?

Given information:

an Euler circuit.

Calculation:

K_n is a simple graph with n vertices v1,v2,...,vn and an edge between every pair of vertices.

An Euler circuit exists when the graph is connected and when every vertex of the graph has an even degree.

K_n is a connected graph, because there is an edge between every pair of vertices. Since each vertex is connected to the other n − 1 vertices by an edge, the degree of every vertex v_i is n − 1

(b)

For what values of n does the complete graph Kn with n vertices have a Hamiltonian circuit?