Question

determine whether the graph has an euler path, an euler circuit, or neither. if the graph has an euler path or circuit, use trial and error or fleurys algorithm to find one. a. choose the correct answer below. the graph has \bigcirc the graph has an euler circuit. \bigcirc the graph has an euler path (but not an euler circuit) \bigcirc the graph has neither an euler path nor an euler circuit.

Explanation:

Step1: Recall the degree - vertex rules

An Euler circuit exists if and only if all vertices in the graph have even degrees. An Euler path exists if and only if exactly two vertices have odd degrees.

Step2: Calculate the degree of each vertex

• Degree of \(A\): \(2\) (connected to \(B\) and \(C\))
• Degree of \(B\): \(2\) (connected to \(A\) and \(C\))
• Degree of \(C\): \(4\) (connected to \(A\), \(B\), \(D\), \(H\))
• Degree of \(D\): \(3\) (connected to \(E\), \(F\), \(C\))
• Degree of \(E\): \(2\) (connected to \(D\) and \(F\))
• Degree of \(F\): \(3\) (connected to \(E\), \(G\), \(D\))
• Degree of \(G\): \(1\) (connected to \(F\))
• Degree of \(H\): \(2\) (connected to \(C\) and \(I\))
• Degree of \(I\): \(1\) (connected to \(H\))

Vertices \(G\) and \(I\) have degree \(1\) (odd), vertices \(D\) and \(F\) have degree \(3\) (odd). So there are \(4\) vertices with odd degrees.

Answer:

The graph has neither an Euler path nor an Euler circuit.