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 exactly two even vertices.
b. it has exactly two odd vertices.
c. it has more than two odd vertices
d. all graphs have at least one euler path.

Explanation:

According to Euler's theorem, a graph has an Euler path if and only if it is connected and has exactly 0 or 2 odd - degree vertices. In this case, we check the degree of each vertex.

• Vertex \(A\): degree \(3\) (odd)
• Vertex \(B\): degree \(4\) (even)
• Vertex \(C\): degree \(3\) (odd)
• Vertex \(D\): degree \(2\) (even)
• Vertex \(E\): degree \(2\) (even)

Since the graph is connected and has exactly two odd - degree vertices (\(A\) and \(C\)), it has an Euler path.

For part \(b\), one possible Euler path (using trial and error) is \(A
ightarrow B
ightarrow C
ightarrow E
ightarrow B
ightarrow D
ightarrow A
ightarrow E\)

Answer:

a. B. It has exactly two odd vertices
b. \(A
ightarrow B
ightarrow C
ightarrow E
ightarrow B
ightarrow D
ightarrow A
ightarrow E\) (one of the possible answers)