Research on Knight's Circuit with Irregular Moves on Square Boards

This paper is about the research on knight's circuits with irregular moves (r,s), r≥1, s2, on square boards. Knight's circuit means the knight moves to every square on the board exactly once and then back to the start, what is a Hamilton circuit. First the paper gives the results already made on this field, then we proof that there is no knight's circuit on a board equal to or smaller than (r+s+1)×(r+s+1). Afterwards the results of practical research are presented to show that there are knight's circuits on boards 2(r+s)×2(r+s). At last we make the supposition that there exists a knight's circuit on boards n×n with n≥2 (r+s) and proof by the method of connecting boards that it is true for a knight with move (r=1,s=4).