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Definition: An edge-component is a sequence of some consecutive collinear-segments.

Consider an $n \times n$ grid-like arrangement of $2n$ lines. Is there any idea about the number of simple cycles with exactly $2n$ edge-components such that each line in the grid contains exactly one edge-component of the cycle? In other words, the cycles should traverse all the lines. A simple cycle is a cycle which passes through the nodes exactly once.

In the picture you can see a $7\times7$ grid-like arrangement and a desired 14-cycle. enter image description here

Any hint or observation is helpful.

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    How about non-simple? If you allow intersections, it might be easier...and you'll also need to know how many 'units' the path traverses.2011-04-26
  • 0
    Then the problem is rather easy, and I know the solution and the exact number. Removing the intersections is worthy.2011-04-27

2 Answers 2