What's this kind of visualization of the Knight's Tour called? Connected rectangular graphs?

Question

What's this kind of visualization of the Knight's Tour called? Connected rectangular graphs? This is on a 6x6 chessboard.

Also, why are the rectangular regions separated like that?

And why does one get similar (3x3) rectangular areas?

Answer 1

This isn't really a visualization of a knight's tour, but a visualization of valid moves from one square to another. Call the upper left corner $A1$ and the lower right $F6$. A knight can hop along the graph lines. A knight's tour would be some path through these nodes that visits each node exactly once.

Each $3 \times 3$ square covers one-quarter of the board, but not geometrically. The set of nodes in each $3 \times 3$ square on the graph comes from successively rotating the board $90$ degrees and applying the pattern in the "$A1$" square on the graph, but with the new upper-left corner:

$$ \begin{array}{c|lcr} & A & B & C & D & E & F\\ \hline 1 & 1 & 2 & 4 & 3 & 1 & 2 \\ 2 & 4 & 3 & 1 & 2 & 4 & 3 \\ 3 & 2 & 1 & 3 & 4 & 2 & 1 \\ 4 & 3 & 4 & 2 & 1 & 3 & 4 \\ 5 & 1 & 2 & 4 & 3 & 1 & 2 \\ 6 & 4 & 3 & 1 & 2 & 4 & 3 \\ \end{array} $$

By splitting things up this way, we can more easily see the fourfold symmetry, as well as see the number of possible moves more easily (from $2$ on the corner squares to $8$ in the four center squares).

Also interesting is that there are no squares with next hops into all four sections. There are no moves that get you from section $1$ to section $3$ directly, for example.