An Eulerian cycle is a directed cycle that contains each edge exactly once. Write a graph client Euler that finds an Eulerian cycle or reports that no such tour exists. Hint : Prove that a digraph G has a directed Eulerian cycle if and only if G is connected and each vertex has its indegree equal to its outdegree.

Square graph. The square of a directed graph G = (V,E) is the graph G^2 =(V,E^2)suchthat(u,w)∈E^2 if and only iff or some v∈V both(u,v)∈E and (v, w) ∈ E; that is, there is a path of exactly two edges. (i) Suppose that G is represented by an adjacency list. Describe an efficient algorithm for computing G^2 from G. (ii) Suppose that G is represented by an adjacency matrix. Describe an efficient algorithm for computing G^2 from G.

An Eulerian cycle is a unidirectional cycle in which each edge appears precisely once. Create a graph client Euler that either discovers an Eulerian cycle or reports that none exist. Prove that a digraph G has a directed Eulerian cycle if and only if it is connected and each vertex's indegree equals its outdegree.

Prove that in any simple graph G (i.e. no multiple edges, no loops) there exist two distinct vertices with the same degrees.