The Knight's Tour Research Paper

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The knight’s tour

In this paper, I will discuss the knight’s tour, a chess puzzle relatable to graph theory. I will talk about the history of the problem, how it is related to the Hamiltonian paths and circuits, and some techniques to finding the many different tours and proving their existence.

The knight is, as you might know, the only chess piece that does not move in a straight line. No, the knight moves two spaces in one direction, and then on in a perpendicular direction. The knights tour is a small journey, where the knight will have to stand on each space, but only once. Figure A: Legal moves for a knight
The knight’s tour can be viewed as a Hamiltonian path, if you consider it a graph. In that case, the vertices would represent
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Proof for (i): If we look at a knight’s moves, we see that every legal move is from a black to a white square, or vice-versa. This means that any closed tour must visit an equal number of white and black squares. The problem is that if both m and n are odd, there will be an odd number of squares, and the knight cannot move to an equal number of black and white squares. This means that it will be impossible to construct a closed tour in a board of odd size.

Proof for (ii) If m = 1 or 2, you can clearly see that the board is not wide enough to construct a tour. And when m = 4, we can prove that there cannot be a knight’s tour using graph coloring. Two different colorings of a 4x6 board are shown in figure C. Figure C: A 4x6 graph.
What is important to notice here is that every time you move from a red square, you can only land on a blue square, but the opposite is not the case if you move from a blue square. From the way the knight moves, we know that since there is no one-to-one correspondence between red/blue squares, it will not be possible to perform a closed
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