Prove the following theorem of Euler.If an undirected graph has more than two vertices of odd degree, it does nothave an Euler path.Hint: Try a proof by contradiction, and model your argument on the proof of Theorem 2.8. **Theorem 2.8** If a graph \( G \) has an Euler circuit, then all the vertices of \( G \) have even degree.**Proof:** Let \( v \) be a vertex in \( G \). If the degree of \( v \) is zero, there is nothing to prove. If the degree of \( v \) is nonzero, then some edges on the Euler circuit touch it, and since the Euler circuit contains all of \( G \)'s edges, the number of touches by these edges equals the degree of \( v \). If we traverse the circuit, starting and ending at \( v \), we must leave \( v \) the same number of times as we return to \( v \). Therefore the number of touches by an edge in the Euler circuit is even.Euler’s observations about paths have related proofs. These are left as exercises.

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If an undirected graph has more than two vertices of odd degree, it does not have an Euler path.

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Question

Prove the following theorem of Euler.
If an undirected graph has more than two vertices of odd degree, it does not
have an Euler path.

Hint: Try a proof by contradiction, and model your argument on the proof of Theorem 2.8.

**Theorem 2.8** If a graph \( G \) has an Euler circuit, then all the vertices of \( G \) have even degree.

**Proof:** Let \( v \) be a vertex in \( G \). If the degree of \( v \) is zero, there is nothing to prove. If the degree of \( v \) is nonzero, then some edges on the Euler circuit touch it, and since the Euler circuit contains all of \( G \)'s edges, the number of touches by these edges equals the degree of \( v \). If we traverse the circuit, starting and ending at \( v \), we must leave \( v \) the same number of times as we return to \( v \). Therefore the number of touches by an edge in the Euler circuit is even.

Euler’s observations about paths have related proofs. These are left as exercises.

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