Decide whether the connected graph has an Euler path, an Euler circuit, an Euler circuit butnot an
Euler path, or neither an Euler circuit nor an Euler path.
The graph has 4 odd vertices and 2 even vertices.
A)Euler path
B)Euler circuit
C)neither an Euler circuit nor an Euler path
D)Euler circuit but not an Euler path

Question

Quizplus · Accepted Answer

A graph with an Euler circuit must have all even vertices, while a graph with an Euler path must have exactly 2 odd vertices (or 0 odd vertices if it is a connected graph). Since this graph has 4 odd vertices, it cannot have an Euler path, and since it has 2 even vertices, it cannot have an Euler circuit. Therefore, the answer is neither an Euler circuit nor an Euler path. A connected graph with an even number of vertices has an Euler circuit if and only if each vertex has even degree. In this case, since there are 4 odd vertices, there is no Euler circuit. Similarly, a connected graph has an Euler path if and only if it has zero or two vertices with odd degree. Since there are 4 odd vertices in this graph, there is no Euler path either.

Decide Whether the Connected Graph Has an Euler Path, an Euler