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Uniform distribution for a class of k-paradoxical oriented graphs

Uniform distribution for a class of k-paradoxical oriented graphs. Joint work with undergraduate students J. C. Schroeder and D. J. Pleshinger (Ohio Northern University, 2012).

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Uniform distribution for a class of k-paradoxical oriented graphs

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  1. Uniform distribution for a class of k-paradoxical oriented graphs Joint work with undergraduate students J. C. Schroeder and D. J. Pleshinger (Ohio Northern University, 2012) With many thanks to the Miami University Fall Conference (2012), where the chosen topic was “Statistics in Sports”. Not being a statistician and definitely wanting to attend and present something, I had to find a suitable topic on short notice. I thought that distribution properties of the dominating sets in the Paley tournaments might not be too far from the conference theme (although some may beg to differ ). Eventually two Ohio Northern University seniors who like to visualize and draw nice pictures of graphs joined me in this investigation, and presented at the 2013 Joint Meetings.

  2. Objects of interest: oriented graphs (no 2 – cycles, no loops, no multiple edges). Notation:

  3. 1 – paradoxical 2 – paradoxical Featured in the paper "On a problem in graph theory“ By P. Erdős (The Mathematical Gazette (1963) 47: 220–223)

  4. A Paley tournament on p = 11 vertices Source: www.ams.jhu.edu/~leslie/paley.eps‎

  5. Paley tournament on p = 23 vertices

  6. Existence of k – paradoxical tournaments: If k is fixed, then large enough n, most tournaments on n vertices are k –paradoxical. Erdős (1963): Non-constructive, probabilistic proof. Sketch of proof

  7. Graham and Spencer (1971) used Paley tournaments G(p) and Weil estimates to provide explicit examples of k – paradoxical tournaments. Proof - preliminaries

  8. Proof

  9. Global symmetry: L/R symmetric if p is of the form 4k+1, anti-symmetric if p is of the form 4k+3. Cumulative sums, p = 17489

  10. Cumulative sums, p = 17491

  11. Proof

  12. A closer look: fragment of a 2-paradoxical oriented graph with 67 vertices. Outgoing edges from x to x+1, x+9, x+14, x+15, x+22, x+24, x+25, x+40, x+59, x+62, x+64 for any x modulo 67. For better visibility, only the outgoing edges from vertices 0,1,…,19 are shown, with the ones emerging from vertex 0 marked in red. This is a subgraph of the Paley tournament G(67) , with one-third the number of edges.

  13. LC: 587, 2777, 1109, 59, 191, 691, 467, 373, 37, 569, 17497, 31649, 1259, 7039, 2243 LC: 419, 353, 167, 10271, 631667, 251, 359, 22079, 2237, 593, 521, 433, 2663, 6551, 5519, 10949, 2371, 563, 277, 4259, 2543, 1009, 71, 397, 109

  14. We found four distinct GPF-Tribonacci limit cycles, of lengths 100, 212, 28 and 6 Maximum cycle element: 18964967822676015504193 Logarithmic plot

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