Question

examine the graph to the right
a. determine whether the graph has an euler path, an euler circuit, or neither.
b. if the graph has an euler path or circuit, use trial and error or fleury’s algorithm to find
one

a. choose the correct answer below
a. the graph has an euler circuit.
b. the graph has an euler path (but not an euler circuit).
c. the graph has neither an euler path nor an euler circuit.
b. if the graph has an euler path or an euler circuit, which of the choices below correctly describes that path or circuit starting at c?
a. c, b, a, b, d, a
b. c, a, d, b, a, b, c
c. c, b, d, a, c
d. the graph has neither an euler path nor an euler circuit.

Explanation:

Step1: Recall Euler circuit and path conditions

A graph has an Euler circuit if all vertices have even degrees. A graph has an Euler path if exactly two vertices have odd degrees.

Step2: Calculate vertex degrees

• Degree of \(A\): \(4\) (even)
• Degree of \(B\): \(4\) (even)
• Degree of \(C\): \(2\) (even)
• Degree of \(D\): \(2\) (even)

Since all vertices have even degrees, the graph has an Euler circuit.

Step3: Check option for Euler circuit starting at \(C\)

Option A: \(C,B,A,B,D,A\) is not a circuit (doesn't end at \(C\))
Option B: \(C,A,D,B,A,B,C\) is a circuit. Check if it uses all edges:

• From \(C\) to \(A\): one edge
• \(A\) to \(D\): one edge
• \(D\) to \(B\): one edge
• \(B\) to \(A\): two edges (used twice)
• \(A\) to \(B\): two edges (used twice)
• \(B\) to \(C\): one edge. All edges are used.

Option C: \(C,B,D,A,C\) is not a circuit (uses only 4 edges, total edges are 6)

Answer:

a. A. The graph has an Euler circuit.
b. B. \(C,A,D,B,A,B,C\)