True or false,
- C5 has a subgraph isomorphic to C4.
- K5 has a subgraph isomorphic to K2,3.
- The complement of C6 is bipartite.
Don't know what C5 etc. indicate.
To determine the answers to these questions, we need to understand some concepts related to graphs and subgraphs.
1. C5 has a subgraph isomorphic to C4 - False:
- C5 refers to a cycle with 5 vertices, which means it has 5 vertices connected in a circular manner.
- C4 refers to a cycle with 4 vertices.
- Since C5 has more vertices than C4, it cannot have a subgraph isomorphic to C4. Therefore, the statement is false.
2. K5 has a subgraph isomorphic to K2,3 - True:
- K5 refers to a complete graph with 5 vertices, where every vertex is connected to every other vertex.
- K2,3 refers to a bipartite graph with 2 vertices in one part and 3 vertices in another part, where vertices in one part are not connected to each other, and vertices in another part are also not connected to each other.
- K2,3 can be visualized as a graph with two isolated vertices on one side and three isolated vertices on the other side.
- Since K5 contains 5 vertices, we can select any 2 vertices from one part and any 3 vertices from another part to form a subgraph isomorphic to K2,3. Therefore, the statement is true.
3. The complement of C6 is bipartite - True:
- The complement of a graph refers to a new graph where the edges that were not present in the original graph are now present, and vice versa.
- C6 refers to a cycle with 6 vertices.
- To find the complement of C6, we need to add all the edges that were not present in C6 and remove the edges that were present in C6.
- The complement of C6 is a bipartite graph.
- A bipartite graph is a graph that can be partitioned into two sets such that vertices within each set are not connected to each other, but vertices from one set are connected to vertices from the other set.
- Since the complement of C6 is bipartite, the statement is true.
In conclusion:
- C5 does not have a subgraph isomorphic to C4 (False).
- K5 does have a subgraph isomorphic to K2,3 (True).
- The complement of C6 is bipartite (True).