Understanding Hamilton Paths and Circuits in Complete Graphs

The video tutorial explores Hamilton paths and circuits in complete graphs K_n for n >= 3. It defines Hamilton paths as walks visiting each vertex once and Hamilton circuits as walks returning to the starting vertex. Examples are provided for K3, K4, K5, and K6, demonstrating the existence of Hamilton paths and circuits. The tutorial concludes that for n >= 3, every complete graph K_n contains a Hamilton path and circuit, as each graph has a cycle on n vertices as a subgraph.

What is a Hamilton path in a graph?

• A path that visits each vertex exactly once

• A path that visits each vertex at least twice

• A path that starts and ends at the same vertex

• A path that visits each edge exactly once

In the context of complete graphs, what is a Hamilton circuit?

• A circuit that does not return to the starting vertex

• A circuit that visits each vertex at least twice

• A circuit that visits each vertex exactly once and returns to the starting vertex

• A circuit that visits each edge exactly once

For the complete graph K3, which of the following is true?

• It has a Hamilton circuit but no Hamilton path

• It has both a Hamilton path and a Hamilton circuit

• It has a Hamilton path but no Hamilton circuit

• It has no Hamilton path

In K4, how can a Hamilton circuit be formed?

• By visiting only half of the vertices

• By visiting each vertex at least twice

• By visiting each edge exactly once

• By visiting each vertex exactly once and returning to the starting vertex

What is a characteristic of a Hamilton path in K4?

• It visits each vertex exactly once

• It visits each edge exactly once

• It starts and ends at the same vertex

• It skips at least one vertex

For K5, what is true about Hamilton paths and circuits?

• It has a Hamilton path but no Hamilton circuit

• It has a Hamilton circuit but no Hamilton path

• It has both a Hamilton path and a Hamilton circuit

• It has no Hamilton path

In K6, how can a Hamilton path be identified?

• By skipping at least one vertex

• By starting and ending at the same vertex

• By visiting each edge exactly once

• By visiting each vertex exactly once

What is a common feature of Hamilton circuits in complete graphs with n vertices?

• They do not return to the starting vertex

• They visit each vertex exactly once and return to the starting vertex

• They visit each edge exactly once

• They visit each vertex at least twice

For n greater than or equal to three, what can be said about complete graphs?

• They have Hamilton circuits but no Hamilton paths

• They have both Hamilton paths and Hamilton circuits

• They have Hamilton paths but no Hamilton circuits

• They have no Hamilton paths

What is the significance of Cn as a subgraph in complete graphs?

• It ensures the graph has a Hamilton path and circuit

• It ensures the graph has no Hamilton circuit

• It ensures the graph has only a Hamilton path

• It ensures the graph has no Hamilton path