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

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Presentation transcript:

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

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

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

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.

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

 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?

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

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

Complete Bipartite with Two Odd Groups

Complete Bipartite with One Odd Group and One Even Group

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

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

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

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