1. Friendly Partitions of Graphs Jenna Huston Advised by Dr. David Offner

2. Graphs  Vertices – white dots  Edges – black lines  Neighbors – Vertices that are connected to another by an edge  A is neighbors with B and D, but not C  Partition - dividing the vertices of a graph into two sets A B C D

3. What is a friendly partition?  Separation of vertices into two groups  Each vertex must have at least as many neighbors in its own group as the other

4. Terms  Happy – the vertex has at least as many neighbors in its side of the partition as the other  Unhappy – the vertex has more neighbors on the other side.

5. Our Question:  What families of graphs have friendly partitions?

6.  Complete Graphs  3-Cycle  Stars  Complete Bipartite with an odd number of vertices in one or both parts. What graphs do not have friendly partitions?

7. Complete Graphs  Each vertex is connected to every other vertex  Equal amount of vertices in each set  Every vertex has one less vertex on its side to connect to than on the other side

8. Complete Graphs  Unequal amount of vertices in each group  The side with fewer is unhappy  There are more connections to the other side

9. Complete Bipartite with Two Odd Groups

10. Complete Bipartite with One Odd Group and One Even Group

11. What graphs have friendly partitions?  k-Cycles where k > 3  Trees with a path of length ≥ 3  Non-connected  Complete Bipartite with an even number of vertices in both parts  Complete k-partite with all groupings having an even number of vertices

12. k-Cycle where k > 3  Place two adjacent vertices on side one of the partition  All others on the other side k is evenk is odd

13. Complete Bipartite with Two Even Groupings  Half of group one is with half of group two

14. Conclusion and Future Work  To investigate more families of graphs for example regular bipartite