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On Hamiltonian cycles in Cayley graphs of wreath products
75
Discrete Mathematics 65 (1987) 75-80 North-Holland
ON HAMILTONIAN OF WREATH
CYCLES IN CAYLEY GRAPHS
PRODUCTS
Richard STONG Department of Mathematics, Harvard University, Cambridge, MA 02138, U.S.A. Received 24 August 1983 Revised 24 June 1986 In this paper we will show that a Hamiltonian cycle exists in the undirected Cayley graph of the wreath product, Z,. wr Z~, of two cyclic groups if m is even or equal to 3.
1. Introduction
Recently there has been increasing interest in fmding Hamiltonian circuits in Cayley digraphs of products of groups with their standard generating sets. Trotter and Erd6s [9], rediscovering an old result of R.A. Rankin [7], found necessary and sufficient conditions on m and n so that the Cayley digraph, Cay(Z~ × Zn), is Hamiltonian. Penn and Witte [6] extended this to determine when Cay(Z,. x Zn) is hypohamiltonian. Curran and Witte [1] proved that Cay(Z~, x Zn~ x - - - x Znk) is Hamiltonian when k I> 3. Witte, Letzter and GaUian [12] showed that Cay(G x Zn) is Hamiltonian when G is dihedral, semidihedral or dicyclic, Letzter [5] considered the same problem when G is metacyclic. Klerlein and Starling [4] obtained partial results for the Cayley digraph of the semidirect product Z,. ~Zn. Keating [3] investigated the Cayley digraphs formed by taking a conjunctive product. Another important group theoretic product is the wreath product; however, the size and complexity of the digraph Cay(Z,. wr Zn) make the search for Hamiltonian circuits difficult. (Independently, Witte and Penn solved the case where n - 2). To simplify this problem, we will consider the undirected Cayley graph of Z,. and Z~. Define the wreath product, Z~ wr Zn, as follows: Let Z,. = (a), Z~-(b), (Zm)n=Z~xZ~x'"xZ,. (n copies of Z,.) and A = (a, O, 0 , . . . , 0)E (Z~) n. Then using the natural action of b on an element (al, a 2 , . . . , an) Of ( Z m ) n gives (an, a l , . . . , an-x). T h i s action leads to a natural semidirect product (Z,.)n ~¢Z~ ffi Z~ wr Z~, which has two natural generators,
(A, 0) and (0, b). The
Cayley
graph,
F({(A, 0), (0, b)): Z~ wr Z~),
henceforth
denoted
/~°)(Z~ wr Zn) has vertex set { ( a l , . . •, an; b): ai e Zm, b e Z~). There is an edge
from (al, a2, • • •, a~;b) to (at, a ~ , . . . , a';b') if and only if either ai=a" for all i and b' - b + 1 or b - b', a~- a" for all i ~ b and a~ - ab ± 1. Let FCk)(Z,. wr Zn) 0012-365X/87/$3.50 ~) 1987, Elsevier Science Publishers B.V. (North-Holland)
R. Stong
76 'l
-(I
Q--
•
n i l
! i
_-/I
I
-
•
'11 i l l
-_([---. :[
l
~.
-
II
m
m
I
i~
fl
g
I
i! ![ IIlIHilli lillillllt I
|lli|lHtHI
~--V-
-It
illlml Ililllllll
iiiiiiii
III!111111 I I I I i i l II!I !
+_
Illi '
•
o,
Illl Illll
:t
m
'D
i
I
Fig: 1. F({(A, 0), (0, b)): Z3wr Za)= F(°)(Z3 wr Z3).
denote the graph with vertex set { ( a k + l , . . . , a,; b): a i e Z m , b e Zn} and an edge from ( a ~ + l , . . . , an; b) to (a~,+l,..., a'; b ' ) if and only if there is an edge in /'<°)(Zm wr Z,,) from ( 0 , . . . +, O , ak+~, . . . , a,,;+b) tO ( 0 , . . . , O, a'k+l, . . . , a ~ , ; b ) , Notice that l ~ k ) ( Z m w r Z , , ) ~ be naturally identified with an. induced subgraph of/~°)(Z,, wr Zn). In group theoretic t e r m i n o t ~ l"(k)(Zm Wr Zn)'is obtained from F
On Hamiltonian cycles in Cayley graphs of wreath products
77
Q--
a t
l
~
(I
•
/~°)(z3wrz2)
r~l)(z3 wr z~)
/t
r~2)(z~ ~r z3)
/~'(z3 wr z2) Fig. 2. Assorted graphs.
m
~
- -
_
.%
i
n T . .
B
, I 11.o .
m Q
m
Q
~
Q
t
.
.
.
.
;
~
~ Q
I . . . . .
•
.
.
.
.
9----
:'1 ~
- 8-
-
i
!
f
~
::
d
-.
:I .:[
•
....
| a
Fig. 3. The cycles mentioned in the proof of Lemma 2 for F°)(Z2 wr Z3) and FO)(Z4 wr 7_.3). Solid lines denote the cycle.
The m is even case Lemma 1.
ff k >12 and F(k)(Z~ wr Zj,) has a Hamiltonian cycle, then so does
~ , - 2 ) ( ~ wr z~). Proof. Let C be a Hamiltonian cycle in/~k). We may assume C starts and ends at
78
R. Stong
the vertex (0, 0 , . . . , 0; k). Note that this vertex has only two neighbors in F (k), namely ( 0 , . . . , 0; k + 1) .and ( 0 , . . . , 0; k - 1). Hence we may assume C has the form C = ( 0 , . . . , 0;k), ( 0 , . . . , 0 ; k + 1 ) , . . . , ( 0 , . . . , 0; k - 1), ( 0 , . . . , 0;k). Let D denote the sequence of (2t)"-kn vertices obtained from C by deleting the last vertex. L e t / 5 denote the reverse of D. Let D(x, y) and D(x, y) denote the natural images of D and /5 in the subgraph of F (k-2) induced by the set { ( a k - 1 , a k , a k + l , . . . , a n ; b ) E i ' ( k - 2 ) : a k - l = X and a k = y } . Therefore the path D(x, y) starts at (x, y, 0 , . . . , 0; k) and ends at (x, y, 0 , . . . , 0; k - 1); the path /)(x, y) starts at (x, y, 0 , . . . , 0; k - 1) and ends at (x, y, 0 , . . . , 0; k.) Define ECv) to be the following sequence of (2t)"-k+t(2n) points:
D(O,y),
/)(1, y),
D(E,y+l),...,D(2t-l,y÷Et-1),
D(l,y+l),
/5(0, y + 2 t - 1).
E(y) is a path which starts at (0, y, 0 , . . . , 0; k) and ends at (0, y + 2 t - 1, 0,...,0;k). The E(2), ( 0 , 0 , . . . , 0 ; k ) .
Hamiltonian []
cycle
is
E(0),
E(-2),
E(-4),...,
Lemma 1 sets up a convenient induction
37aeorem 2. F(°)(Zetwr Z,) has a Hamiltonian cycle. Proof. Observe that F(")(Z~wrZ.) is cyclic hence has a Hamiltonian cycle. Therefore for n even Lemma 1 gives a Hamiltonian cycle in Ft°)(Z2~ wr Z,). If n is odd Lemma 1 gives a Hamiltonian path in F("-e)(Z2~ wr Z,) whose first n vertices are (0, 0; n), (0, 0; 1 ) , . . . , (0, 0; n - 1) and whose last n vertices are (0, 1; n - 1), (0, 1, n - 2 ) , . . . , (0, 1; n). Let Co denote the natural image of this path in the subgraph of Ft"-3)(Z2t wr Z,) with first components equal to a. Let (~= be the path obtained from Co by replacing its first n vertices and last n vertices as follows. Replace the first n vertices of Co with the sequence (a, 0, 0; n - 2), (a, 0, 0; n - 3 ) , . . . , (a, 0, 0, 1), (a, 0, 0; n), (a, 0, 0; n - 1). Replace the last n vertices of Co with the sequence (a, 0, 1; n - 1), (a, 0, 1; n), (a, 0, 1; 1 ) , , . . , (a, 0, 1; n - 2). Observe that (~ is still a Hamiltonian path in the subgraph of
-a o -._: - ;
i:::i t.....
. . . . . . . .
~,
:
,j
--j
Fig. 4. The cycle mentioned in F(1)(Z2 wr Z4). Solid lines denote the cycle.
On Hamiltonian cycles in Cayley graphs of wreath products
79
F in-3) with first components equal to a. Let t~'a denote the reverse of t~o. The sequence t~lt~t~3(7~---C2~_1C~(1, O, 0;n) is a Hamiltonian cycle in /~n-3). Lemma 1 gives a cycle in ~o). []
3. The m eqaai to 3 case
For the case when m = 3 we will make use of the cyclic, normal subgroup N = (((a, a , . . . , a), 0)) of Zm wr Zn. We will call a path in Ftk)(z3 wr Z,,) "good" if it contains exactly one element of each coset of N, that is if the path contains exactly one element of the form (ak+~ + c , . . . , an + c; b) for 0 ~
Proof. For k = n - 1 the path is (0, 0; n - 2), (0, 0; n - 1 ) , . . . , (0, 0; n), (0, 0; n - 1), (1, 0; n - 1),
(2, 0;n - 1),(2, 0;n - 2 ) , . . . , (2, 0;n), (2, 1;n), (2, 1; 1), . . . , (2, 1 ; n - 2 ) . For k < n - 1 we proceed by induction. Let ( 0 , . . . , 0 ; k - I ) , P, ( 2 , . . . , 2, 1; k - 1) denote the good path in Fk-I(Za wr Zn). Let Po denote the image of P in / " ( / - 2 ) ( Z 3 wr Zn) with first coordinate a. Let Pa denote Pa in the reverse direction. Then P0, (0, 2 , . . . , 2, 1 ; k - 1), (2, 2 , . . . , 2, 1 ; k - 1), (1, 2 , . . . , 2, 1 ; k - 1), P1, (1, 0 , . . . , 0 ; k - 1), (0, 0 , . . . , 0 ; k - 1), (2, 0 , . . . , 0 ; k - 1), P2 is the good path in Ftk-2)(Z3 wr Z~).
[]
I~eorem 4. Ft°)(Z3 wr Zn) has a Hamiltonian cycle.
Proof. Lemma 3 gives a good path in/~°)(Z a wr Zn) beginning at (0, 0 , . . . , 0; n) and ending at (2, 2 , . . . , 2, 1; n). Denote this path by P and let po) denote the path obtained by adding (1, 1 , . . . , 1; 0) componentwise to each vertex of P, similarly let p(2) denote the path obtained by adding (2, 2 , . . . , 2;0) componentwise to each vertex of P. Then P, p(2), p(1), (0, 0 , . . . , 0; n) is a Hamiltonian cycle in Ft°)(Z~ wr Zn). [] For m odd and not equal to 3 the endpoints of the paths constructed in this manner are not separated by an edge.
R. Stong
80
4. The n equal to 2 for digraphs case David Witte and Larry Penn, independently, noticed that the directed graph Cay({(A, 0), (0, b)}: Zm wr Z2) has a Hamiltonian circuit, hence /~°)(Zm wr Z2) has a Hamiltonian cycle. Let h(x) denote the path: (x, x; 1), (x + 1, x; 1 ) , . . . ,
(x - 1, x; 1), (x - 1, x; 2), (x - 1, x + 1; 2),
. . . , ( x - 1, x - 1;2). Then the sequence h ( n ) , h ( n - 1 ) , . . . , h(1), (0, 0; 1) is the Hamiltonian circuit.
Acknowledgments This work was done at a summer research program sponsored by the University of Minnesota at Duluth. I would like to thank Professor Joseph A. Gallian, who ran the program and suggested this problem, and David Witte, who on several occasions provided helpful advice.
Re/erences [1] S. Curran and D. Witte, Hamiltonian paths in cartesian products of directed cycles, Ann. Discrete Math., to appear. [2] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969). [3] K. Kcating, The conjunction of Cayley digraphs, Discrete Math. 42 (1982) 209-219. [4] J.B. Klerlein and A.G. Starling, Hamiltonian cycles in Cayley color graphs of semi-direct products, Proc. 9th SE Conf. Combinatorics, Graph Theory and Computing, Congress, Numer.
xxi (1976) 411-435. [5] G. Letzter, Hamiltonian circuits in cartesian products with a metaeyclic factor, Ann. Discrete Math., to appear. [6] L. Penn and D. Witte, When the cartesian product of two directed cycles is hypohamiltonian, J. Graph Theory, 7 (1983) 441-443. [7] R.A. Rankin, a companological problem in group theory, Proc. Camp. Phil. Soc. 44 (1948) 17-25. [8] W.R. Scott, Group Theory, (Prentice-Hall, Englewood Cliffs, NJ, 1964). [9] W.T. Trotter and P. ErdOs, When the cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137-142. [10] D. WiRe, On Hamiltonian circuits in Cayley diagrams, Discrete Math. 38 (1982) 99-108. [11] D. Witte, Personal communication. [12] D. Witte, G. Letzter and J. GaUian, Hamiltonian circuits in cartesian products of Cayley digraphs, Discrete Math. 43 (1983) 297-307.
Discrete Mathematics 65 (1987) 75-80 North-Holland
ON HAMILTONIAN OF WREATH
CYCLES IN CAYLEY GRAPHS
PRODUCTS
Richard STONG Department of Mathematics, Harvard University, Cambridge, MA 02138, U.S.A. Received 24 August 1983 Revised 24 June 1986 In this paper we will show that a Hamiltonian cycle exists in the undirected Cayley graph of the wreath product, Z,. wr Z~, of two cyclic groups if m is even or equal to 3.
1. Introduction
Recently there has been increasing interest in fmding Hamiltonian circuits in Cayley digraphs of products of groups with their standard generating sets. Trotter and Erd6s [9], rediscovering an old result of R.A. Rankin [7], found necessary and sufficient conditions on m and n so that the Cayley digraph, Cay(Z~ × Zn), is Hamiltonian. Penn and Witte [6] extended this to determine when Cay(Z,. x Zn) is hypohamiltonian. Curran and Witte [1] proved that Cay(Z~, x Zn~ x - - - x Znk) is Hamiltonian when k I> 3. Witte, Letzter and GaUian [12] showed that Cay(G x Zn) is Hamiltonian when G is dihedral, semidihedral or dicyclic, Letzter [5] considered the same problem when G is metacyclic. Klerlein and Starling [4] obtained partial results for the Cayley digraph of the semidirect product Z,. ~Zn. Keating [3] investigated the Cayley digraphs formed by taking a conjunctive product. Another important group theoretic product is the wreath product; however, the size and complexity of the digraph Cay(Z,. wr Zn) make the search for Hamiltonian circuits difficult. (Independently, Witte and Penn solved the case where n - 2). To simplify this problem, we will consider the undirected Cayley graph of Z,. and Z~. Define the wreath product, Z~ wr Zn, as follows: Let Z,. = (a), Z~-(b), (Zm)n=Z~xZ~x'"xZ,. (n copies of Z,.) and A = (a, O, 0 , . . . , 0)E (Z~) n. Then using the natural action of b on an element (al, a 2 , . . . , an) Of ( Z m ) n gives (an, a l , . . . , an-x). T h i s action leads to a natural semidirect product (Z,.)n ~¢Z~ ffi Z~ wr Z~, which has two natural generators,
(A, 0) and (0, b). The
Cayley
graph,
F({(A, 0), (0, b)): Z~ wr Z~),
henceforth
denoted
/~°)(Z~ wr Zn) has vertex set { ( a l , . . •, an; b): ai e Zm, b e Z~). There is an edge
from (al, a2, • • •, a~;b) to (at, a ~ , . . . , a';b') if and only if either ai=a" for all i and b' - b + 1 or b - b', a~- a" for all i ~ b and a~ - ab ± 1. Let FCk)(Z,. wr Zn) 0012-365X/87/$3.50 ~) 1987, Elsevier Science Publishers B.V. (North-Holland)
R. Stong
76 'l
-(I
Q--
•
n i l
! i
_-/I
I
-
•
'11 i l l
-_([---. :[
l
~.
-
II
m
m
I
i~
fl
g
I
i! ![ IIlIHilli lillillllt I
|lli|lHtHI
~--V-
-It
illlml Ililllllll
iiiiiiii
III!111111 I I I I i i l II!I !
+_
Illi '
•
o,
Illl Illll
:t
m
'D
i
I
Fig: 1. F({(A, 0), (0, b)): Z3wr Za)= F(°)(Z3 wr Z3).
denote the graph with vertex set { ( a k + l , . . . , a,; b): a i e Z m , b e Zn} and an edge from ( a ~ + l , . . . , an; b) to (a~,+l,..., a'; b ' ) if and only if there is an edge in /'<°)(Zm wr Z,,) from ( 0 , . . . +, O , ak+~, . . . , a,,;+b) tO ( 0 , . . . , O, a'k+l, . . . , a ~ , ; b ) , Notice that l ~ k ) ( Z m w r Z , , ) ~ be naturally identified with an. induced subgraph of/~°)(Z,, wr Zn). In group theoretic t e r m i n o t ~ l"(k)(Zm Wr Zn)'is obtained from F
On Hamiltonian cycles in Cayley graphs of wreath products
77
Q--
a t
l
~
(I
•
/~°)(z3wrz2)
r~l)(z3 wr z~)
/t
r~2)(z~ ~r z3)
/~'(z3 wr z2) Fig. 2. Assorted graphs.
m
~
- -
_
.%
i
n T . .
B
, I 11.o .
m Q
m
Q
~
Q
t
.
.
.
.
;
~
~ Q
I . . . . .
•
.
.
.
.
9----
:'1 ~
- 8-
-
i
!
f
~
::
d
-.
:I .:[
•
....
| a
Fig. 3. The cycles mentioned in the proof of Lemma 2 for F°)(Z2 wr Z3) and FO)(Z4 wr 7_.3). Solid lines denote the cycle.
The m is even case Lemma 1.
ff k >12 and F(k)(Z~ wr Zj,) has a Hamiltonian cycle, then so does
~ , - 2 ) ( ~ wr z~). Proof. Let C be a Hamiltonian cycle in/~k). We may assume C starts and ends at
78
R. Stong
the vertex (0, 0 , . . . , 0; k). Note that this vertex has only two neighbors in F (k), namely ( 0 , . . . , 0; k + 1) .and ( 0 , . . . , 0; k - 1). Hence we may assume C has the form C = ( 0 , . . . , 0;k), ( 0 , . . . , 0 ; k + 1 ) , . . . , ( 0 , . . . , 0; k - 1), ( 0 , . . . , 0;k). Let D denote the sequence of (2t)"-kn vertices obtained from C by deleting the last vertex. L e t / 5 denote the reverse of D. Let D(x, y) and D(x, y) denote the natural images of D and /5 in the subgraph of F (k-2) induced by the set { ( a k - 1 , a k , a k + l , . . . , a n ; b ) E i ' ( k - 2 ) : a k - l = X and a k = y } . Therefore the path D(x, y) starts at (x, y, 0 , . . . , 0; k) and ends at (x, y, 0 , . . . , 0; k - 1); the path /)(x, y) starts at (x, y, 0 , . . . , 0; k - 1) and ends at (x, y, 0 , . . . , 0; k.) Define ECv) to be the following sequence of (2t)"-k+t(2n) points:
D(O,y),
/)(1, y),
D(E,y+l),...,D(2t-l,y÷Et-1),
D(l,y+l),
/5(0, y + 2 t - 1).
E(y) is a path which starts at (0, y, 0 , . . . , 0; k) and ends at (0, y + 2 t - 1, 0,...,0;k). The E(2), ( 0 , 0 , . . . , 0 ; k ) .
Hamiltonian []
cycle
is
E(0),
E(-2),
E(-4),...,
Lemma 1 sets up a convenient induction
37aeorem 2. F(°)(Zetwr Z,) has a Hamiltonian cycle. Proof. Observe that F(")(Z~wrZ.) is cyclic hence has a Hamiltonian cycle. Therefore for n even Lemma 1 gives a Hamiltonian cycle in Ft°)(Z2~ wr Z,). If n is odd Lemma 1 gives a Hamiltonian path in F("-e)(Z2~ wr Z,) whose first n vertices are (0, 0; n), (0, 0; 1 ) , . . . , (0, 0; n - 1) and whose last n vertices are (0, 1; n - 1), (0, 1, n - 2 ) , . . . , (0, 1; n). Let Co denote the natural image of this path in the subgraph of Ft"-3)(Z2t wr Z,) with first components equal to a. Let (~= be the path obtained from Co by replacing its first n vertices and last n vertices as follows. Replace the first n vertices of Co with the sequence (a, 0, 0; n - 2), (a, 0, 0; n - 3 ) , . . . , (a, 0, 0, 1), (a, 0, 0; n), (a, 0, 0; n - 1). Replace the last n vertices of Co with the sequence (a, 0, 1; n - 1), (a, 0, 1; n), (a, 0, 1; 1 ) , , . . , (a, 0, 1; n - 2). Observe that (~ is still a Hamiltonian path in the subgraph of
-a o -._: - ;
i:::i t.....
. . . . . . . .
~,
:
,j
--j
Fig. 4. The cycle mentioned in F(1)(Z2 wr Z4). Solid lines denote the cycle.
On Hamiltonian cycles in Cayley graphs of wreath products
79
F in-3) with first components equal to a. Let t~'a denote the reverse of t~o. The sequence t~lt~t~3(7~---C2~_1C~(1, O, 0;n) is a Hamiltonian cycle in /~n-3). Lemma 1 gives a cycle in ~o). []
3. The m eqaai to 3 case
For the case when m = 3 we will make use of the cyclic, normal subgroup N = (((a, a , . . . , a), 0)) of Zm wr Zn. We will call a path in Ftk)(z3 wr Z,,) "good" if it contains exactly one element of each coset of N, that is if the path contains exactly one element of the form (ak+~ + c , . . . , an + c; b) for 0 ~
Proof. For k = n - 1 the path is (0, 0; n - 2), (0, 0; n - 1 ) , . . . , (0, 0; n), (0, 0; n - 1), (1, 0; n - 1),
(2, 0;n - 1),(2, 0;n - 2 ) , . . . , (2, 0;n), (2, 1;n), (2, 1; 1), . . . , (2, 1 ; n - 2 ) . For k < n - 1 we proceed by induction. Let ( 0 , . . . , 0 ; k - I ) , P, ( 2 , . . . , 2, 1; k - 1) denote the good path in Fk-I(Za wr Zn). Let Po denote the image of P in / " ( / - 2 ) ( Z 3 wr Zn) with first coordinate a. Let Pa denote Pa in the reverse direction. Then P0, (0, 2 , . . . , 2, 1 ; k - 1), (2, 2 , . . . , 2, 1 ; k - 1), (1, 2 , . . . , 2, 1 ; k - 1), P1, (1, 0 , . . . , 0 ; k - 1), (0, 0 , . . . , 0 ; k - 1), (2, 0 , . . . , 0 ; k - 1), P2 is the good path in Ftk-2)(Z3 wr Z~).
[]
I~eorem 4. Ft°)(Z3 wr Zn) has a Hamiltonian cycle.
Proof. Lemma 3 gives a good path in/~°)(Z a wr Zn) beginning at (0, 0 , . . . , 0; n) and ending at (2, 2 , . . . , 2, 1; n). Denote this path by P and let po) denote the path obtained by adding (1, 1 , . . . , 1; 0) componentwise to each vertex of P, similarly let p(2) denote the path obtained by adding (2, 2 , . . . , 2;0) componentwise to each vertex of P. Then P, p(2), p(1), (0, 0 , . . . , 0; n) is a Hamiltonian cycle in Ft°)(Z~ wr Zn). [] For m odd and not equal to 3 the endpoints of the paths constructed in this manner are not separated by an edge.
R. Stong
80
4. The n equal to 2 for digraphs case David Witte and Larry Penn, independently, noticed that the directed graph Cay({(A, 0), (0, b)}: Zm wr Z2) has a Hamiltonian circuit, hence /~°)(Zm wr Z2) has a Hamiltonian cycle. Let h(x) denote the path: (x, x; 1), (x + 1, x; 1 ) , . . . ,
(x - 1, x; 1), (x - 1, x; 2), (x - 1, x + 1; 2),
. . . , ( x - 1, x - 1;2). Then the sequence h ( n ) , h ( n - 1 ) , . . . , h(1), (0, 0; 1) is the Hamiltonian circuit.
Acknowledgments This work was done at a summer research program sponsored by the University of Minnesota at Duluth. I would like to thank Professor Joseph A. Gallian, who ran the program and suggested this problem, and David Witte, who on several occasions provided helpful advice.
Re/erences [1] S. Curran and D. Witte, Hamiltonian paths in cartesian products of directed cycles, Ann. Discrete Math., to appear. [2] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969). [3] K. Kcating, The conjunction of Cayley digraphs, Discrete Math. 42 (1982) 209-219. [4] J.B. Klerlein and A.G. Starling, Hamiltonian cycles in Cayley color graphs of semi-direct products, Proc. 9th SE Conf. Combinatorics, Graph Theory and Computing, Congress, Numer.
xxi (1976) 411-435. [5] G. Letzter, Hamiltonian circuits in cartesian products with a metaeyclic factor, Ann. Discrete Math., to appear. [6] L. Penn and D. Witte, When the cartesian product of two directed cycles is hypohamiltonian, J. Graph Theory, 7 (1983) 441-443. [7] R.A. Rankin, a companological problem in group theory, Proc. Camp. Phil. Soc. 44 (1948) 17-25. [8] W.R. Scott, Group Theory, (Prentice-Hall, Englewood Cliffs, NJ, 1964). [9] W.T. Trotter and P. ErdOs, When the cartesian product of directed cycles is Hamiltonian, J. Graph Theory 2 (1978) 137-142. [10] D. WiRe, On Hamiltonian circuits in Cayley diagrams, Discrete Math. 38 (1982) 99-108. [11] D. Witte, Personal communication. [12] D. Witte, G. Letzter and J. GaUian, Hamiltonian circuits in cartesian products of Cayley digraphs, Discrete Math. 43 (1983) 297-307.