Question

Question 2: Prove that a connected graph has an Euler path if and only if it has exactly two vertices of odd degree.

Answer 1

Prove that a connected graph has an Euler path if and only if it has exactly two vertices of odd degree.

Explanation:

if a Graph has vertices with even degree the Graph  will contain a Euler Circuit.Ee Let G contain a Euler path.  A graph contains a Euler path if it can travel each vertex but doesn't end up on the identical vertex.  There is an additional edge (That makes two vertices have an odd degree).

Let G have two vertices that have odd degree and the rest have even degree. Case I: If we start from an even degree vertex we will be trapped on a vertex on the odd degree vertex as ultimately we will have an edge to travel to the vertex but no edge to travel away from it.