Consider two integers n, m > 1. Prove that Kn, m admits an Eulerian circuit if and only if both n and m are even.
Recall that:
1. a trail is a walk without repeated edges,
2. an Eulerian trail is a trail passing through every edge, and
3. a graph admits an Eulerian trail if and only if the following hold (a) every vertex has even degree or there are exactly two vertices with odd degree and (b) the vertices of nonzero degree are connected.
Note that the following algorithm may be used to construct Eulerian trails.
• Step 1. – If every vertex has even degree then pick any vertex v of nonzero degree and construct a circuit from v to itself. – If there are exactly two vertices with odd degrees then construct a trail between them.
• Step 2. As long as some vertex w belongs to the current trail and has incident edges not in the current trail, construct another circuit from w to itself using only previously unused edge and join it to the current circuit.
ANSWER THE FOLLOWING:
- Prove that Kn, m admits a Eulerian circuit if and only if both n and m are even.
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