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.

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Presentation transcript:

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 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?

Extremal Graphs of Higher Order

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

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

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

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

More Advanced Search Hill Climbing with checking – Builds quickly – Results similar to Hill Climbing, but faster

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

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

Our Best Method f(v)

Another Trick Hill Descending – Steps down from known extremal graph

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