Euler Paths and Circuits in Complete Bipartite Graphs

The video tutorial explains Euler paths and circuits in complete bipartite graphs K_m,n. It defines an Euler path as a walk through a graph using every edge once, existing if at most two vertices have odd degrees. An Euler circuit is a closed walk using every edge once, existing if all vertex degrees are even. The tutorial examines vertex degrees in various bipartite graphs and analyzes three cases: both m and n even, both odd, and one even and one odd. It concludes with conditions under which Euler paths and circuits exist in these graphs.

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

• The graph must be a tree.

• The graph must be connected.

• There must be exactly two vertices with odd degree.

• All vertices must have even degree.

In a complete bipartite graph K3,3, what is the degree of each vertex?

• 5

• 3

• 2

• 4

What is the degree of each vertex in the graph K2,2?

• 4

• 3

• 2

• 1

In the graph K3,4, what is the degree of the vertices in the larger partition?

• 5

• 4

• 3

• 6

If both m and n are even in a complete bipartite graph K_m,n, what can be said about Euler paths and circuits?

• Neither an Euler path nor an Euler circuit exists.

• There is an Euler path but no Euler circuit.

• There is an Euler circuit but no Euler path.

• There are both an Euler path and an Euler circuit.

Which of the following graphs has an Euler circuit?

• K3,3

• K2,3

• K2,2

• K1,2

What is the exception for the existence of an Euler path in the case where both m and n are odd?

• K3,3 has an Euler circuit.

• K3,3 has an Euler path.

• K1,1 has an Euler path.

• K1,1 has an Euler circuit.

For a complete bipartite graph K_m,n, if m and n are both odd, what is true?

• There is an Euler path.

• There is both an Euler path and an Euler circuit.

• There is an Euler circuit.

• There are neither an Euler path nor an Euler circuit.

In the case where m is odd and n is even, under what condition does K_m,n have an Euler path?

• When m and n are both odd.

• When m or n is 1.

• When m or n is 2.

• When m and n are both even.

If m is even and n is odd in K_m,n, when does the graph have an Euler path?

• When m or n is 1.

• When m or n is 2.

• When m and n are both even.

• When m and n are both odd.