mad_19b03

What is an Euler circuit in a graph?

• A simple path containing every vertex of the graph

• A simple path that avoids repeating any vertex

• A simple circuit containing every edge of the graph

• A circuit that contains all nodes at least once

Which of the following is NOT a characteristic of an Euler path?

• It contains every edge of the graph

• • It must return to the starting point

• It can start and end at different vertices

• It includes every edge exactly once

According to the image, which graph contains an Euler cycle?

• Graph with all vertices having even degree

• Graph with all vertices having odd degree

• Graph with two vertices having odd degree

• Graph with more than four vertices

In the historical origin of the Euler cycle, why was it impossible to cross each bridge once and return to the starting point?

• The bridges were too far apart

• The graph had vertices with odd degree

• The bridges were not connected

• There were too many vertices in the graph

Which of the following graphs does NOT contain an Euler circuit?

• A graph with all even degree vertices

• A graph with one odd degree vertex

• A graph with three odd degree vertices

• A graph with two odd degree vertices

What is a necessary condition for a graph to have an Euler circuit?

• The graph must have exactly two vertices with odd degrees.

• The graph must be disconnected.

• All vertices must have even degrees.

• The graph must have at least one isolated vertex.

What is the degree of a vertex?

• • The number of edges in the graph.

• • The number of edges connected to that vertex.

• The number of paths in the graph.

• • The number of vertices adjacent to that vertex.

Which graph has an Euler circuit?

• A graph with one vertex of odd degree.

• A graph with two vertices of odd degree.

• A graph where all vertices have even degrees.

• A graph where all vertices have odd degrees.

What is an Euler path?

• A path that passes through every edge of the graph exactly once and returns to the starting point.

• A path that passes through every edge of the graph exactly once but doesn't return to the starting point.

• A path that passes through every vertex exactly once.

• A path that passes through every vertex at least once.

In the graph on the left in "Theorem 1," which vertex has the highest degree?

• Vertex a

• Vertex b

• Vertex c

• Vertex d