Hamilton Circuit Review

In a Euler Circuit, we visit:

• Every vertex.

• Every edge.

• Only the starting point.

• Only the endpoints.

What is a Euler Circuit?

• A path in a graph that visits every edge exactly once and returns to the starting vertex.

• A path that visits every vertex in a graph exactly once.

• A closed path that visits every vertex at least once.

• A path that connects all vertices with the minimum possible total edge weight.

What is a complete graph?

• A graph where no vertices are connected by edges.

• A graph in which every pair of distinct vertices is connected by a unique edge.

• A graph that contains only one vertex.

• A graph that is disconnected and has no edges.

Can a graph have a Hamilton Circuit if it has isolated vertices?

• Yes, a graph can have a Hamilton Circuit with isolated vertices.

• No, a graph with isolated vertices cannot have a Hamilton Circuit.

• Only if the isolated vertices are connected by edges.

• It depends on the number of isolated vertices.

What is the significance of the degree of vertices in Hamilton Circuits?

• Each vertex must have a degree of at least 2 to ensure that it can be visited in a Hamilton Circuit.

• Each vertex must have a degree of exactly 1 to form a Hamilton Circuit.

• Each vertex can have any degree as long as the total number of vertices is even.

• Each vertex must have a degree of at least 3 to ensure a Hamilton Circuit exists.

What is a path in graph theory?

• A path is a sequence of edges that connects a sequence of vertices.

• A path is a single vertex in a graph.

• A path is a collection of isolated vertices.

• A path is a loop that connects back to the starting vertex.

How many Hamilton Circuits does a complete graph with 4 vertices have?

• 4

• 5

• 6

• 7

What is the minimum degree of each vertex in a Hamilton Circuit?

• At least 1.

• At least 2.

• At least 3.

• At least 4.

What is the difference between a Hamilton Circuit and a Euler Circuit?

• A Hamilton Circuit visits every vertex exactly once, while a Euler Circuit visits every edge exactly once.

• A Hamilton Circuit visits every edge exactly once, while a Euler Circuit visits every vertex exactly once.

• Both circuits visit every vertex and edge exactly once.

• A Hamilton Circuit can visit vertices multiple times, while a Euler Circuit cannot.

How do you determine if a graph has a Hamilton Circuit?

• There is a simple algorithm that can be applied directly.

• It often requires checking various paths or using heuristics.

• You can always find a Hamilton Circuit by trial and error.

• Graphs with an even number of vertices always have a Hamilton Circuit.