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Unicursal Circuit

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A unicursal circuit is a circuit that traverses every graph edge of a graph exactly once. Examples of curves that can be traced unicursally are the Mohammed sign and unicursal hexagram.

For n>=3 fixed points in cyclic order on a circle, let a(n) count closed polygonal routes that visit each point exactly once and do not join neighboring points on the circle. Routes are identified under a change of starting point or reversal of direction, while the positions on the circle remain fixed. With the conventions a(1)=1 and a(2)=0, the numbers of these unicursal polygrams 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).

For n>=8, this sequence also satisfies 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 Resource. https://mathworld.wolfram.com/UnicursalCircuit.html

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