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Unicursal Circuit

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A circuit in which an entire graph is traversed in one route. Examples of curves that can be traced unicursally are the Mohammed sign and unicursal hexagram.

The numbers of distinct unicursal polygrams that can be formed from n points on circle with no two adjacent for n=1, 2, ... are 1, 0, 0, 0, 1, 3, 23, 177, 1553, ... (OEIS A002816). For n>2, these are given by the sum

a(n)=(-1)^n+1/2(n-1)!+sum_(k=1)^(n-1)sum_(j=1)^k((-1)^k2^(j-1)(k-1; j-1)(n-k; j)n(n-k-1)!)/(n-k).

This sequence is also given by the recurrence equation

(n^2-7n+9)a(n)=(n-5)(n^2-5n+3)a(n-5)+(n^2-7n+9)a(n-4)-2(n-6)(n^2-5n+3)a(n-3) +(n^3-8n^2+18n-21)a(n-1)+4(n-5)na(n-2).

See also

Eulerian Cycle, Graph Cycle, Hexagram, Königsberg Bridge Problem, Mohammed Sign, Traceable Graph

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References

Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 223-224, 1930.Sloane, N. J. A. Sequence A002816/M3102 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 256-257, 1999.

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Unicursal Circuit

Cite this as:

Weisstein, Eric W. "Unicursal Circuit." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/UnicursalCircuit.html

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