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 an euler path (but not an euler circuit)
the graph has an euler circuit
the graph has neither an euler path nor an euler circuit

Explanation:

Step1: Count the degree of each vertex

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

Step2: Apply Euler's theorem

Euler's theorem states that a graph has an Euler circuit if and only if all vertices have even degrees. A graph has an Euler path if and only if exactly two vertices have odd degrees. Here, vertices \(A\), \(C\), \(E\), \(F\) have odd degrees (\(4\) vertices with odd degrees).

Answer:

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