an elementary technique to construct Hamilton cycles in a finite graph G on which a cyclic ... for no element of Za but 0 fixes each r-errex and edge of Gn.r. ..... I.J. Dejter, Equiaalenl condilioni for Euler's Problenr on14-Hamillon Cgcles, Ars ...
SCIENTIA Series A : Malhemalical Sciences, Vol.
(1990 /1991), 21-29
Universidad Tdcnica Federico Santa trIaria Valparaiso, Chile
Quarter-T\rrns and Hamiltonian Cycles
for Annular Chessknight Graphs. Italo J. Dejter Abstract. In this article, we discuss some aspects of the search for chessknight closed tours 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
in a
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 Z- acts 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 Z- concatenate 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.
1. lntroduction. Consider the euclidean plane I furnished as usual with orthogonal cartesian coordinates, that allow us to identify X u'ith R?, where R is the real line. For i : 0, 1 let zr; : D -+ R be the standard projection onto the itlt-coordinate. We think of the square chessboards as centered at the origin in I and with their sides respectively parallel to the coordinate axes. Graph vertices are taken to be the chess entry centers. Then, if n is an integer larger than 1, the n x n-chessboard is identified with the vertex
