I’ve written several times about knight’s tours of a chessboard. The paths in these tours cross each other many times. What if you wanted to look tours that do not cross themselves? You can’t reach every square this way. You can reach half of them, but no more than half.
The following tour is part of the article Uncrossed Knight’s Tours in Donald Knuth’s book Selected Papers on Fun & Games.
The longest non-crossing knight path on an 8×8 board visits 35 squares, which is more than half of the board.
If you mean a *closed* non-crossing knight path, the claim is also false for many board sizes (the smallest being 9×9).
A tour is a closed path.