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The Search for Extremal Graphs

The Search for Extremal Graphs. Richard Ligo, Aaron Zavora, and Stephen Donnel Advised by Dr. David Offner. What is a graph?. e = the number of edges v = the number of vertices C 3 is a 3-cycle C 4 is a 4-cycle. The Search for Extremal Graphs.

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The Search for Extremal Graphs

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  1. The Search for Extremal Graphs Richard Ligo, Aaron Zavora, and Stephen Donnel Advised by Dr. David Offner

  2. What is a graph? • e = the number of edges • v = the number of vertices • C3 is a 3-cycle • C4 is a 4-cycle

  3. The Search for Extremal Graphs Our Question: Given a graph with v vertices, what is greatest number of edges, f(v), the graph can have without containing a 3 or 4-cycle?

  4. Extremal Graphs of Higher Order

  5. Previous Work • Conjecture: (Erdős 1975) • Known theoretical bounds: • For all v: • For infinitely many v: (Garnick, Kwong, Lazebnik, 1993)

  6. Previous Work By: David K. Garnick, Y. H. Harris Kwong, Felix Lazebnik

  7. Hills

  8. Computer Search for Extremal Graphs • Hill Climbing • Most basic method • No backtracking (you are never “descending”) • Results match the results found by Garnick et. al. for v < 36

  9. Previous Work By: David K. Garnick, Y. H. Harris Kwong, Felix Lazebnik

  10. More Advanced Search • Hill Climbing with checking • Builds quickly • Results similar to Hill Climbing, but faster

  11. Previous Work By: David K. Garnick, Y. H. Harris Kwong, Felix Lazebnik

  12. Our Best Method • Hill Building • Builds on known extremal graphs • Surpasses the results found by Garnick, Kwong, and Lazebnik for v = 45, 55, 57, 58, 59, 60, 75, 76, 81, 82, 85, 86, 87

  13. Our Best Method

  14. Another Trick • Hill Descending • Steps down from known extremal graph

  15. Future Areas of Exploration • Improve computer results • Improve current Hill Climbing and Building Strategies • Implementation new search strategies • Try to prove a given graph is extremal • Apply methods to other problems

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