When the cartesian product of directed cycles is Hamiltonian

The cartesian product of two hamiltonian graphs is always hamiltonian . For directed graphs, the analogous statement is false . We show that the cartesian product C, x C„ 2 of directed cycles is hamiltonian if and only if the greatest common divisor (g .c .d .) d of n, and n 2 is at least two and there exist positive integers d,, d 2 so that d, + d2 = d and g.c.d . (n,, d,) = g .c.d. (n 2, d2 ) = 1 . We also discuss some number-theoretic problems motivated by this result . 1 . INTRODUCTION Let G, = (V,, E,) and G2 = (V2 , E,) he graphs . The cartesian product (see p. 22 of [1]) of G, and G 2 , denoted G, x G 2 , is the graph G = (V, E) where V = V, x V2 and and E _ {{(is,, v,), (u2, v2)} u,=u2 or u, = v 2 and William T. Trotter, Jr . UNIVERSITY OF SOUTH CAROLINA Paul Erdös HUNGARIAN ACADEMY OF SCIENCES {v,, v2}e E2 u u EE A graph G = ( V, E) is hamiltonian if there exists a listing v,, v 2 V" of the vertex set V so that {v ;, v, ., ,} E E for i = 1, 2, . . . , n -1 and {v,,, v,} E E. It is elementary to show that if G, and G 2 are hamiltonian, so is G,xG2 . Now let G, = ( V,, E,) and G 2 = ( V2 , E2 ) he directed graphs, i .e, E, and Journal of Graph Theory, Vol . 2 (1978) 137-142 @ 1978 by John Wiley & Sons, Inc . 0364-9024/78/0002-0137$01 .00 138 JOURNAL OF GRAPH THEORY E Z are sets of ordered pairs of V, and V 2 , respectively . The cartesian product G, X GZ is the directed graph G = (V, E) where V = V, X VZ, and E _ 1((u', vi), (UZ, v2)) or U,= u2 and (v,, v 2) E EZ v, = v 2 and (u,, UZ) E E, A directed graph G = (V, E) is said to be hamiltonian if there exists a listing v, v 2 , . . . , vn of V so that (v,, v, + ,) E E for i _ 1, 2, . . ., nI and (vn , v,) E E. As we shall see, the cartesian product of hamiltonian directed graphs need not be hamiltonian .