Does Warnsdorff’s algorithm for Knight’s tour problem always gives complete tour
In general Warnsdorff's rule is just a heuristic that guides the search. It is still possible that the search hits a dead-end and we are forced to backtrack.

So let us consider the $n \times n$ chessboard now.
Warnsdorff's rule (nor any other method) won't find a solution for $n &lt; 5$ as a solution exists precisely when $n \geq 5$. Given that $n \geq 5$, an efficient algorithm based on Warnsdorff's rule is described in [1, Proposition 2.2]. Other independently discovered algorithms that work in linear time are known as well (see e.g., [2]).



[1] Conrad, Axel, Tanja Hindrichs, Hussein Morsy, and Ingo Wegener. "Solution of the knight's Hamiltonian path problem on chessboards." Discrete Applied Mathematics 50, no. 2 (1994): 125-134.

[2] Parberry, Ian. "An efficient algorithm for the Knight's tour problem." Discrete Applied Mathematics 73, no. 3 (1997): 251-260.