Discrete Maths Graph theory

Question

What are the necessary and sufficient conditions for Euler path and Circuit in directed graph?

Answer 1

Euler Path:

1.If the graph is connected and has exactly 2 odd degree vertices then there exists at least one Euler Path.Mainly it starts with one end and ends on the other End.

Euler Circuit:

1.If the graph is connected and every vertex has even degree then the Graph has atleast one Euler Circuit.

Euler Circuit is a part of Euler Path but Converse is nt True.

# of ODD Vertices	Implication (for a connected graph)
0	There is at least  one Euler Circuit.
1	THIS IS IMPOSSIBLE!
2	There is no Euler Circuit but at least 1 Euler Path.
more than 2	There are no Euler Circuits or Euler Paths.

Comments

These are for undirected graph,right? I need for directed graph

---

Eulerian Path is a path in graph that visits every edge exactly once. Eulerian Circuit is an Eulerian Path which starts and ends on the same vertex.

A directed graph has an eulerian cycle if following conditions are true. 
1) All vertices with nonzero degree belong to a single strongly connected component.
2) In degree and out degree of every vertex is same.

Answer 2

Euler Path -->> If a graph is weakly connected or is w/o repetition of edges and

Indegree= outdegree for all except for two vertices. One out of those two has indegree = outdegree+1. Other has  indegree+1 = outdegree.

Euler Circuit -->>   If a graph is weakly connected or is w/o repetition of edges and

Indegree= outdegree.