Question

use the graph to determine whether the path described is an euler path, an euler circuit, or neither. explain your answer. a, b, e, f, b, c, d, e, c, a choose the correct answer below. a. the path described is an euler path (but not an euler circuit) because it begins and ends at the same vertex. b. the path described is an euler circuit because it travels through every edge of the graph exactly once, and it begins and ends at the same vertex. c. the path described is neither an euler path nor an euler circuit, because the vertices d, e, and c are all odd. d. the path described is an euler path (but not an euler circuit) because it does not begin and end at the same vertex. e. the path described is neither an euler path nor an euler circuit because it begins and ends at the same vertex. f. the path described is an euler circuit because it does not begin and end at the same vertex.

Explanation:

An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is an Euler path that starts and ends at the same vertex.
First, check if the path uses every edge. The path \(A,B,E,F,B,C,D,E,C,A\) does not use all the edges (for example, the edge \(BC\) is used but other edges like \(BD\) are not).
Second, check the vertices' degrees. For a graph to have an Euler path, it must have exactly 0 or 2 vertices of odd degree. For an Euler circuit, all vertices must have even degree.
Counting degrees:

• Degree of \(A\): 2 (edges \(AB\) and \(CA\))
• Degree of \(B\): 4 (edges \(AB\), \(BE\), \(BF\), \(BC\))
• Degree of \(C\): 4 (edges \(BC\), \(CD\), \(CE\), \(CA\))
• Degree of \(D\): 1 (edge \(CD\))
• Degree of \(E\): 3 (edges \(BE\), \(FE\), \(CE\))
• Degree of \(F\): 1 (edge \(BF\))

Since vertices \(D\) and \(F\) have degree 1 (odd) and \(E\) has degree 3 (odd), there are more than 2 vertices of odd degree. Also, the path does not use all edges.

Answer:

C. The path described is neither an Euler path nor an Euler circuit because the vertices \(D\), \(E\), and \(C\) are all odd.