Question

a graph is given to the right.
a. explain why the graph has at least one euler path.
b. use trial and error or fleury’s algorithm to find one such path.
a. choose the correct explanation below.
a. it has more than two odd vertices.
b. it has exactly two even vertices.
c. it has exactly two odd vertices.
d. all graphs have at least one euler path.
b. write an euler path starting at a.
(use a comma to separate vertices as needed.)

Explanation:

Step1: Determine vertex degrees

• Degree of \(A\): \(2\) (even)
• Degree of \(B\): \(1\) (odd)
• Degree of \(C\): \(1\) (odd)
• Degree of \(D\): \(2\) (even)
• Degree of \(E\): \(4\) (even)

Step2: Find Euler path

Since the graph has exactly two odd vertices (\(B\) and \(C\)), and we start at \(A\) (an even - degree vertex), we can use trial - error.
Start at \(A\), move to \(B\) (since we need to cover the odd - degree vertex \(B\) eventually). Then from \(B\) to \(E\), \(E\) to \(A\), \(A\) to \(D\), \(D\) to \(E\), \(E\) to \(C\)

Answer:

a. C. It has exactly two odd vertices.
b. \(A,B,E,A,D,E,C\)