Quarter - T \ rrns and Hamiltonian Cycles for Annular Chessknight Graphs

In this article, we discuss some aspects of the search for chessknight closed tours in a chessboard, as in [1], [g], [10], [14] and [16] and give an elementary application to the interaction between graph theory and group theory. These closed tours will be herewith denoted as chessknight Hamilton cycles. In particular, we prove the following: If n and r are integers>0withn4r> 2,thenthedifference,4\Boftwoconcentricsquareboards,4 and I with (n + 2r)2 and n2 entries respectively has a chessknight Hamitton cycle invariant under quarter-turns if and only if r > 2 and either n or r is odd. In proving this, we apply an elementary technique to construct Hamilton cycles in a finite graph G on which a cyclic group Zacts freely, a theme present in the literature (see References) in different periods and contexts. This technique consists in looking for a path in G whose images under the action of Zconcatenate in a cyclic fashion to form a Hamilton cycle. Our treatment of a group acting on a graph is elementary and the reader needs only to be acquainted with the notion of a group G acting on a set S, in which case G is a group of permutations of.9, as treated in some standard algebra textbooks.