Eulerian Graphs Quiz

Which condition must be true for a graph to be Eulerian?

• All vertices with non-zero degree are connected, and exactly two vertices have odd degrees.

• All vertices with non-zero degree are connected, and all vertices have even degrees.

• All vertices must have the same degree.

• The graph must be a complete graph.

What is an Eulerian circuit?

• A path that visits every vertex exactly once.

• A path that visits every edge exactly once and starts and ends at the same vertex.

• A path that visits every edge exactly once but can start and end at different vertices.

• A path that visits some edges more than once.

In a graph with 5 vertices where each vertex is connected to every other vertex, how many edges are there?

• 5

• 10

• 15

• 20

Which statement is true for Eulerian paths?

• An Eulerian path exists if and only if exactly zero vertices have odd degrees.

• An Eulerian path exists if and only if exactly two vertices have odd degrees.

• An Eulerian path exists if and only if all vertices have even degrees.

• An Eulerian path exists if and only if more than two vertices have odd degrees.

Apply Euler's theorem: If a graph has 6 vertices and each vertex has a degree of 4, how many edges does the graph have?

• 10

• 12

• 24

• 6

Which of the following graphs is not Eulerian?

• A graph where all vertices have even degrees.

• A graph with three vertices of degrees 2, 4, and 6.

• A complete graph with five vertices.

• A graph with four vertices, each connected to every other vertex.

What is the minimum number of edges a simple graph with 4 vertices needs to be Eulerian?

• 2

• 3

• 4

• 6

In a graph, if every vertex has an odd degree, which of the following is true?

• The graph is definitely Eulerian.

• The graph has an Eulerian path but not an Eulerian circuit.

• The graph cannot have an Eulerian path or circuit.

• The graph has an Eulerian circuit but not an Eulerian path.

Which of the following is a necessary condition for a graph to have an Eulerian circuit?

• The graph is connected.

• The graph has no cycles.

• Each vertex has an odd degree.

• The graph is bipartite.

• Yes, it has an Eulerian path.

• No, it does not have an Eulerian path.

• Yes, it has an Eulerian circuit.

• No, it does not have an Eulerian circuit.