Prove that a connected graph G is an Euler graph if and only if every vertex of G
is of even degree.
Answers
An Eulerian circuit is a traversal of all the edges of a simple graph once and only once, staring at one vertex and ending at the same vertex. You can repeat vertices as many times as you want, but you can never repeat an edge once it is traversed.
The degree of a vertex is the number of edges incident with that vertex.
So let GG be a graph that has an Eulerian circuit. Every time we arrive at a vertex during our traversal of GG, we enter via one edge and exit via another. Thus there must be an even number of edges at every vertex. Therefore, every vertex of GG has even degree.
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Answer:
A connected graph G is an Euler graph if and only if it has an Eulerian cycle, which is a cycle that visits every edge exactly once. This is true if and only if the graph is connected and the degree of each vertex is even.
Suppose all vertices of G have even degree. Then, since the sum of all vertex degrees in a graph is equal to twice the number of edges, the number of edges in the graph must also be even. This means that it is possible to traverse every edge exactly once in a cycle.
On the other hand, if G is an Euler graph, then it has an Eulerian cycle, and this cycle visits every edge exactly once. Thus, every vertex must be of even degree, since an edge joining two vertices adds two to their degree.
Therefore, a connected graph G is an Euler graph if and only if all vertices of G are of even degree