Jun 30, 1998 - A family G of isomorphic copies of a given digraph. ~G is said to be an ... was completely solved by Bennett and Wu 2] and, independently, by Ganter and ... More recently, Granville, Gronau and Mullin 8] proved that for every k.
Orthogonal decompositions of complete digraphs Sven Hartmann FB Mathematik, Universitat Rostock, 18051 Rostock, Germany June 30, 1998
Abstract
~ is said to be an orA family G of isomorphic copies of a given digraph G ~ ~ , if every arc of thogonal decomposition of the complete digraph Dn by G D~ n belongs to exactly two members of G and the union of any two di erent elements from G contains precisely one pair of reverse arcs. Given a digraph H~ , an H~ -family mH~ is the vertex-disjoint union of m copies of H~ . In this paper, we consider orthogonal decompositions by H~ -families. Our objective is to prove the existence of such an orthogonal decomposition whenever certain necessary conditions hold and m is suciently large.
1 Introduction ~ is said to be a decomposition of D ~ i A collection G of subdigraphs of a digraph D ~ every arc of D belongs to exactly one member of G . Throughout, the subdigraphs in G are called the pages of the decomposition. If, in addition, all pages in G are ~ , we speak of a decomposition by G~ . isomorphic to a given digraph G ~ n denote the complete (symmetric) digraph on n For any positive integer n, let D ~ n is said to be orthogonal if the union of any two vertices. A decomposition G of D ~ . distinct pages in G contains exactly one digon, i.e. a copy of D Decomposition problems of graphs have received much attention during the last two decades. Orthogonal decompositions were studied rst by Hering and Rosenfeld [10], ~ n by a directed (n ? 1)-cycle. Even who asked for an orthogonal decomposition of D though this problem is far from being solved, it gave rise to a considerable number of results concerning orthogonal decompositions by special digraphs. Several papers deal with the undirected analogue of the problem, i.e. with orthogonal double covers of complete graphs. For a survey on this topic see Alspach, Heinrich and Liu [1]. For a digraph H~ , let mH~ denote the vertex-disjoint union of m copies of H~ and call it an H~ -family of size m. In this paper, we concentrate on orthogonal decompositions by H~ -families. In the sequel, let H~ always denote a simple digraph, i.e. without multiple arcs, loops and digons. ~ n be V = f0; 1; : : : ; n?1g. A page G~ v is said to be idempotent Let the vertex set of D 2
3
i it contains vertex v as a singleton (or does not contain vertex v at all). An orthogonal decomposition shall be called idempotent i all its pages are idempotent. When considering keys and antikeys in databases, Demetrovics and Katona [6] asked ~ n by directed 3-cycle families. A rst, for idempotent orthogonal decompositons of D partial answer to this question was given by Demetrovics, Furedi and Katona [5]. ~ . Finally, the problem Rausche [12] proved that there is no such decomposition of D was completely solved by Bennett and Wu [2] and, independently, by Ganter and Gronau [7] who proved the existence of an idempotent orthogonal decomposition of D~ n by a directed 3-cycle family i n 1 mod 3 and n 6= 10. More recently, Granville, Gronau and Mullin [8] proved that for every k 3 there ~ n by a directed k-cycle family exists an idempotent orthogonal decomposition of D whenever the necessary condition n 1 mod k holds and n is suciently large. This question was asked in the undirected case by Hering [9]. Our objective is to generalize the latter result to arbitrary digraph families. We shall ~ n by an H~ -family prove the existence of an idempotent orthogonal decomposition of D for almost all n whenever certain necessary conditions hold. These preconditions are given by the following lemma. Lemma 1. Let H~ be a simple digraph with v(H~ ) vertices and e(H~ ) arcs. If there ~ n by an H~ -family then v(H~ ) exists an idempotent orthogonal decomposition of D e(H~ ) and n 1 mod e(H~ ) hold. ~ n by an Proof. Assume, there exists an idempotent orthogonal decomposition of D H~ -family G~ = mH~ of size m. For the numbers of vertices and arcs in D~ n we obtain ~ n) = n v(G~ ) + 1 = m v(H~ ) + 1 v (D and ~ n) = n(n ? 1) = n e(G~ ) = nm e(H~ ); e(D respectively. This implies e(H~ ) v(H~ ) as well as n ? 1 0 mod e(H~ ). The di erence e(H~ ) ? v(H~ ) is usually called the cyclomatic number of the digraph H~ . Thus we are looking for orthogonal decompositions by H~ -families where H~ has non-negative cyclomatic number. 10
2 Preliminaries In this section we shall assemble some terminology and basic results which will be required in the sequel. Throughout, let q denote a prime power, and let a and b be 4
positive integers such that q = ab + 1 holds. Consider V = f0; : : : ; q ? 1g to be the nite eld GF (q) of order q. Let ? denote its cyclic multiplicative group containing all non-zero elements of GF (q), and let T be its unique subgroup of cardinality a and index b. If w denotes a generator of ?, then T is clearly generated by wb. A character on ? is a map from ? to the complex numbers satisfying j (x)j = 1 and (xy) = (x) (y) for all x; y 2 ?. The characters on ? form again a cyclic group of order q ? 1, the dual group ? of ?. It will be convenient to extend the usual de nition of a character on ? by putting (
(0) = 1 if = , 0 otherwise, 0
where is the principal character with (x) = 1 for all x 2 ?. Let T be the unique subgroup of index a and cardinality b in ?. If denotes a generator of ? then T is clearly generated by a. The interplay between the characters in T and the elements in T is given by the following well-known equality (see e.g. [14]): 0
X
2T
(x) =
b?1 X j =0
aj (x) =
8 > : 0 otherwise:
(1)
Let f be a polynomial of positive degree d over GF (q), and let E be an extension eld of GF (q). Then f is said to split in E i f can be written as a product of linear factors over E , i.e. i there exist elements x ; : : : ; xd 2 E such that 1
f (x) = z(x ? x ) (x ? xd); 1
where z is the leading coecient of f . The values x ; : : : ; xd are called the roots of f . Note that there is a (up to isomorphism) unique smallest extension eld containing all roots of f , the splitting eld of f . For more detailed information on polynomials and splitting elds, we refer to [13]. P We are interested in sums of the form x2GF q (f (x)), where is a xed character and f a xed polynomial over GF (q). The problem of evaluating character sums is known to be dicult in general. One usually has to be satis ed with estimates for the absolute value of such sums. We shall make use of the following result [15]. (The interested reader may consult e.g. [13, 14].) Theorem 2 (Weil). Let f be a polynomial of positive degree over GF (q) such that f is not the b-th power of a polynomial over GF (q). Let r be the number of pairwise 1
( )
5
distinct roots of f in a splitting eld of f . Then
j
X
x2GF (q)
a(f (x))j (r ? 1)q 1=2
denotes a generator of the dual group ? .
holds, where
If X and Y are two subsets of ?, their product XY is de ned as the set fxy 2 ? : x 2 X and y 2 Y g. If X = fxg we shall brie y write xY instead of fxgY . Moreover, let X ? denote the set fx? : x 2 X g of all inverses of the elements in X. Let w be a generator of ?. We now consider the cosets Ti = wiT of ? modulo T , where i is some integer. Ti and Tj denote the same coset whenever i j mod b holds. The cosets themselves form again a group ?=T known as the factor group of ? (with respect to T ). The order of ?=T is just (q ? 1)=jT j = b. We continue with a somewhat technical result that forms the foundation of what follows. Lemma 3. Let b be a positive integer and r; s be non-negative integers. There is a constant q = q (b; r; s) such that for all prime powers q q with q 1 mod b, for all r-sets U GF (q), for all s-sets Z GF (q) and for all maps : U ! ?=T there exists an element x 2 GF (q) satisfying x 62 Z; (2) x ? u 2 (u) (3) for every u 2 U . 1
3
1
3
3
Proof. The claim is trivial when r = 0 holds. Conversely, let r be positive. An element x 2 GF (q) lies in coset Ti modulo T i w?i x lies in T , where w denotes as above a generator of ?. For every integer i and for every element x 2 GF (q), put 8 > b if x = 0; Si(x) = b : 2T 0 otherwise. 1
(4)
The idea is to use such sums to construct an expression which has value 1 whenever (3) holds and becomes suciently small otherwise. For the sake of simplicity, we de ne a function : U ! f0; : : : ; b ? 1g given by (u) = T u for every u 2 U . Next, we put Y P (x) = S u (x ? u): ( )
( )
u2U
6
Some consequences of (4) may be noted immediately. We have P (x) = 1 i (3) holds. Moreover, P (x) is positive i x ? u 2 (u) [ f0g holds for every u 2 U . This implies 0 < P (x) < 1 i we have x ? u 2 (u) [ f0g for every u 2 U , and x lies in U . Hence there are at most r elements x satisfying 0 < P (x) < 1. We emphasize that in this case we actually have 0 < P (x) 1=b. Finally, we put X Q= P (x): x2GF (q)
Then Q > r=b implies the existence of an element x 2 GF (q) with P (x) = 1, i.e. (3) holds. The existence of an appropriate x satisfying (2) and (3) will follow if Q > r=b + s. We shall show that Q becomes larger than r=b + s if q is suciently large. To begin with, we nd that X Y X (w? u (x ? u)) Q = b1r x2GF q u2U 2T ( )
( )
X = b1r x2GF q
( )
b?1 YX
u2U j =0
aj (w?(u) (x ? u));
where denotes as above a generator of ?. Let M denote the set of all r-tuples j = (ju : u 2 U ) 2 f0; : : : ; b ? 1gr . On expanding and recombining Q, we obtain X X Y aju (w? u (x ? u)) Q = b1r j 2M x2GF q u2U X X Y a(w?ju u (x ? u)ju ) = b1r j 2M x2GF q u2U X X Y a(w? Pu2U ju u = b1r (x ? u)ju ): j 2M x2GF q u2U ( )
( )
( )
( )
( )
( )
For every tuple j 2 M , put
zj = w?
Pu2U ju u
( )
and let
fj (x) =
Y
u2U
(x ? u)ju
7
;
P
be a polynomial of degree u2U ju over GF (q). If j is the tuple with all zero-entries, simply denoted by o, then zo = w = 1 and fo is the constant polynomial of value 1 for all x 2 GF (q). Hence, we obtain 0
X
x2GF (q)
a(z
o fo (x)) =
X
x2GF (q)
a(1) = q:
(5)
Conversely, let j be di erent from o. The roots of fj are just the elements u 2 U which satisfy ju > 0. Hence, fj has at most r distinct roots. Since all components ju in j are smaller than b, and at least one of them is positive, fj is not the b-th power of a polynomial over GF (q). Applying Weil's theorem, we obtain
j
X
x2GF (q)
a(z
j fj (x))j = j
a(z
j )jj
X
x2GF (q)
a (f
j (x))j (r ? 1)q
=
1 2
(6)
for every j 6= o. Combining (5) and (6) we obtain X X X a(z f (x)) = 1 a (z f (x)) Q ? b1r o o j j r b j 6 o x2GF q x2GF q X X X a(z f (x))j 1 a (z f (x))j jQ ? b1r j o o j j r b j6 o x2GF q x2GF q jQ ? b1r qj b1r (br ? 1)(r ? 1)q = jQ ? b1r qj (r ? 1)q = =
( )
( )
=
( )
( )
1 2
1 2
and thus
Q b1r q ? (r ? 1)q = ; 1 2
which tends to in nity when q ! 1. So x can be chosen satisfying (2) and (3) if q is large enough. This concludes the proof. A straightforward, but tendious calculation shows that q = q (b; r; s) = b r(r + s) + 1 will be a lower bound for q to ensure the existence of an appropriate x 2 GF (q). 2
3
2
3
In the sequel, let k denote the set f1; : : : ; kg. A choice over GF (q) is a map C : k ! ?=T assigning a coset modulo T to each pair (i; j ) of integers i; j 2 k . With the help of our Lemma 3 it is easy to verify that there exists a k-tuple x = 2
8
(x ; : : : ; xk ) of elements in ? satisfying xj ? xi 2 C (i; j ) for every pair i; j 2 k with i < j whenever q is large enough. The latter result is originally due to Wilson [16, 18] and was used to prove the asymptotic existence of pairwise balanced designs [19]. For details, we refer to [3]. Our objective is to prove several generalizations of Wilson's result. Lemma 4. Let b; k be positive integers. There is a constant q = q (b; k) such that for all odd prime powers q q with q 1 mod b and for all maps C : k ! ?=T there exists a k-tuple x of elements from GF (q) satisfying 1
4
4
?xi + xj 2 C (i; j ) xj 2 C (i; j ) xi + xj 2 C (i; j ) for all pairs (i; j ) 2 k .
if i < j; if i = j; if i > j
4
2
(7)
2
Proof. We shall proceed by induction on k. For k = 1 the claim is trivially true: just choose x1 from the coset C (1; 1) which is non-empty for q 2. Assume, the statement is true for k ? 1, and we shall prove it for k, now. Let x = (x1; : : : ; xk?1) be a (k ? 1)-tuple satisfying (7) for each pair (i; j ) 2 2k?1. It remains to nd an element xk 2 C (k; k) with xk ?xi 2 C (i; k) and xk +xi 2 C (k; i) for every i 2 k?1 . It turns out that Lemma 3 ensures the existence of such an element xk when putting Z = ; and U = f0; x1; : : : ; xk?1; ?x1; : : : ; ?xk?1g. Note that the elements 0, xi (i 2 k?1) and ?xi (i 2 k?1) are pairwise distinct by virtue of the induction hypothesis and since q is odd. We must now de ne a map on U by 8 > C (i; k) if u = xi; : C (k; i) if u = ?xi for every element u 2 U . Hence, we are looking for an element xk 2 GF (q) such that xk ? u 2 (u) holds for every u 2 U . By Lemma 3, such an element exists whenever q q (b; 2k ? 1; 0). 3
Unfortunately, for even prime powers, the statement of Lemma 3 is usually false since y ? x = y + x holds for all elements x; y 2 GF (q) if q is even. However, the following weaker result holds.
9
Lemma 5. Let b; k be positive integers. There is a constant q = q (b; k) such that for all even prime powers q q with q 1 mod b and for all maps C : k ! ?=T 5
5
2
5
there exists a k-tuple x of elements from GF (q) satisfying
xi + xj = ?xi + xj 2 C (i; j ) xj 2 C (i; j ) for all pairs (i; j ) 2 k .
if i < j; if i = j
2
Proof. The proof works analogously to the proof of Lemma 4. This time we obtain q = q (b; k; 0). 5
3
We now consider choices C over GF (q) assigning a coset modulo T to every 4-tuple (g; h; i; h) 2 k . Applying Lemma 3 we shall verify the following claim which holds for both, even and odd prime powers. Lemma 6. Let b; k be positive integers. There is a constant q = q (b; k) such that for all prime powers q q with q 1 mod b and for all maps C : k ! ?=T there exists a k-tuple x with elements from GF (q) satisfying 4
6
6
xj 2 C (g; h; i; j ) ?xi + xj 2 C (g; h; i; j ) ?xg xh + xixj 2 C (g; h; i; j )
6
4
if g = h = i = j; if g = h = i < j; (8) if g h < i < j or g < i < h < j or i < g h < j
for all 4-tuples (g; h; i; j ) 2 4k . Proof. Again we shall proceed by induction on k. However, we shall prove the slightly stronger statement that there is a k-tuple x satisfying not only (8) but also
(9) xg xhxi 6= xg0 xh0 xi0 whenever two multisets fg; h; ig and fg0; h0; i0g over k are distinct. When k = 1 the claim trivially holds: just choose x in the coset C (1; 1; 1; 1) which is non-empty for q 2. Assume, the statement is true for k ? 1, and we shall prove it for k, now. Let x = (x ; : : : ; xk? ) be a (k ? 1)-tuple satisfying (8) for each 4-tuple (g; h; i; j ) 2
k? as well as our additional property (9). We are interested in an element xk 2 C (k; k; k; k) with xk ? xi 2 C (i; i; i; k) for every i 2 k? and xk ? xg xhx?i 2 C (g; h; i; k)C (i; i; i;i)? for every 3-tuple (g; h; i) 2 k? , where
k? = f(g; h; i) 2 k? : g h < i or g < i < h or i < g hg: 1
4
1
1
1
+
1
+
1
3
1
10
1
1
1
In order to apply Lemma 3, we put
U = f0; x ; : : : ; xk? g [ fxg xhx?i : (g; h; i) 2 k? g; 1
1
1
+
1
and
Z = XXXX ? X ? [ fz : z 2 XXXX ? g [ fz : z 2 XXX g; 1
1
2
1
3
where X denotes the set fx ; : : : ; xk? g containing the components of the (k ? 1)tuple x. On U we de ne a map : U ! ?=T by 1
1
8 > C (j; j; j; k) : C (g; h; i; k)C (i; i; i;i)?
1
1
for every integer j 2 k? and every 3-tuple (g; h; i) 2 k? . In general, this de nition will only be correct if the elements 0, xj (j 2 k? ) and xg xhx?i ((g; h; i) 2 k? ) are pairwise distinct. It is readily veri ed that the elements 0 and xj (j 2 k? ) are pairwise distinct by the induction hypothesis. For the same reason, all the elements xg xhx?i are di erent from 0. Next suppose, there are two distinct 3-tuples (g; h; i) and (g0; h0; i0) in k? such that xg xhx?i = xg0 xh0 x?i0 holds. This implies xg xhxi0 = xg0 xh0 xi, and by virtue of (9) for k ? 1, the multisets fg; h; i0g and fg0; h0; ig are equal. Since i 6= g; h, we obtain i = i0. Moreover, g h and g0 h0 imply g = g0 as well as h = h0. Clearly, this gives a contradiction. Finally suppose, there is an integer j 2 k? and a 3-tuple (g; h; i) 2 k? such that xj = xgxhx?i holds. This implies xixj = xg xh, and again by the induction hypothesis (9), the multisets fi; j g and fg; hg are equal. But i 6= g; h, which gives a contradiction. By Lemma 3, there exists an element xk 2 GF (q) satisfying xk ? u 2 (u) for every u 2 U and xk 62 Z if q q (b; k ; 6k ). Note, that jU j k and jZ j 6k hold. To satisfy our additional property (9) we have to ensure xg xhxi 6= xg0 xh0 xi0 whenever the multisets fg; h; ig and fg0; h0; i0g over k are distinct. If k is neither in fg; h; ig nor in fg0; h0; i0g, then the claim follows from the induction hypothesis. Conversely assume, there are two distinct multisets fg; h; ig and fg0; h0; i0g over
such that xgxhxi = xg0 xh0 xi0 holds and k lies in fg; h; i; g0; h0; i0g. Thus we obtain xk 2 XXXX ? X ? or xk 2 XXXX ? or xk 2 XXX . But such a relation contradicts xk 62 Z . This concludes the proof. +
1
1
+
1
1
1
1
1
+
1
1
+
1
1
3
1
1
3
5
3
2
1
11
3
5
1
1
The preceding lemmas require large prime powers q satisfying the additional condition q 1 mod b for a given integer b. The existence of such prime powers is ensured by Dirichlet's well-known theorem on primes in arithmetic progressions (cf. [11]). We record this result for future reference. Theorem 7 (Dirichlet). Let p ; b; r be positive integers with g:c:d:fb; rg = 1. There exists a prime p p such that p r mod b holds. 7
7
By a further investigation we may render the following slightly modi ed existence result on primes in arithmetic progressions. Lemma 8. Let p ; b; r be positive integers with g:c:d:f2b; rg = 1. There exist two primes p; q with q > p p such that p q r mod b and g:c:d:fp(p?r); q(q ?r)g = 2b hold. 8
8
Proof. By Dirichlet's theorem there is a prime p p8 with p 4b + r mod 2b(2b + r), say p = 2b(2b + r) + 4b + r for some positive integer . Put = (2b + r) + 2, such that p = 2b + r holds. Again by Dirichlet's theorem there is a prime q > p with q 2b + r mod 2bp, say q = 2bp + 2b + r for some positive integer . Clearly, p q r mod b holds. Moreover, we have g:c:d:fp(p ? r); q(q ? r)g = g:c:d:fp(p ? r); q ? rg = g:c:d:f2bp; 2bp + 2bg = 2b g:c:d:fp; p + 1g = 2b:
3 Main constructions ~ n admits For a given digraph H~ , let IH~ denote the set of all integers n such that D an idempotent orthogonal decomposition by an H~ -family. Our ultimative aim in this section is to investigate the occurency of prime powers in IH~ . In order to study ~ q where q is a prime power, we orthogonal decompositions of complete digraphs D shall exploit the concepts presented in Section 2. It is advantageous to distinguish several cases. To begin with, we consider odd prime powers q satisfying the additional property that (q ? 1)=e(H~ ) is even. Lemma 9. Let H~ be a simple digraph with non-negative cyclomatic number. There ~ ) such that every odd prime power q q with q exists a constant q = q (H 1 mod 2e(H~ ) lies in IH~ . 9
9
9
~ ) and b = e(H~ ). Consider V = f0; : : : ; q ? 1g to be the nite eld Proof. Put k = v(H GF (q) of order q = ab + 1. Throughout, we shall use the terminology introduced in Section 2. In particular, let ? be the multiplicative group of GF (q), and T its 12
unique subgroup of cardinality a and index b. When w denotes a generator of ?, then the cosets modulo T are given by Ti = wiT . ~, Let W = f1; : : : ; kg and B = fal : l = 1; : : : ; bg be the vertex set and arc set of H respectively. On W we de ne a map C : W ! ?=T by 8 > Tl if (i; j ) = al 2 B or (j; i) = al 2 B; : T otherwise: 2
2
1
0
Put q = q (b; k) and let q q : Applying Lemma 4 we nd a k-tuple x satisfying xj 2 Tj? for all vertices j in H~ , and xj ? xi 2 Tl as well as xj + xi 2 Tl for all arcs al = (i; j ) 2 B . It is worth mentioning that both, the di erences xj ? xi and the sums xi + xj taken over all arcs (i; j ) in H~ form a system of representatives of the cosets modulo T . For every v 2 V and for every t 2 T , we construct a digraph H~ v (t) as the isomorphic image of H~ under the map j ! txj + v for all vertices j in H~ . First, we shall show that for every xed v 2 V the digraphs H~ v (t) with t 2 T are pairwise vertex-disjoint. Assume, there are distinct elements t and t0 in T such that the digraphs H~ v (t) and H~ v (t0) share a vertex z. Then there are two vertices j and j 0 in H~ such that z = txj + v = t0xj0 + v holds. This implies txj = t0xj0 . So xj and xj0 lie in the same coset modulo T . Due to our choice of C , we obtain j = j 0 and consequently t = t0, which gives a contradiction. Hence, for xed v 2 V the ~ v . It should digraphs H~ v (t), t 2 T , form an H~ -family aH~ which will be denoted by G be mentioned that G~ v has exactly e(G~ v ) = ae(H~ ) = q ? 1 arcs. Secondly, we point out that G~ v is idempotent for every v 2 V . Assume, v is a vertex in G~ v . This implies the existence of a vertex j in H~ satisfying txj + v = v for some element t 2 T . However, this contradicts xj 6= 0 for all j in H~ . Thus G~ v is idempotent. ~ v , v 2 V , form a decomposition G of D ~ q. Next, we shall check that the digraphs G Let G denote the collection of all digraphs G~ v , v 2 V . Since each of the q digraphs ~ q belongs to in G contains exactly q ? 1 arcs, it suces to show that every arc of D ~ v. at least one of the digraphs G ~ q . In H~ we nd exactly one arc (i; j ) such that Let (y; z) be an arbitrary arc in D the di erence xj ? xi lies in the same coset modulo T as the di erence z ? y. Put t = (z ? y)(xj ? xi)? , and v = y ? txi = z ? txj . Clearly, t is an element of T , and v lies in V . It is now easy to see that the digraph H~ v (t) contains the arc (y; z), and ~ v . We conclude that every arc of D ~ q lies in at least one page of so does the page G G , and consequently, G forms a decomposition of the complete digraph D~ q . 9
4
9
1
1
13
~ v and G~ w be two distinct pages It still remains to prove that G is orthogonal. Let G ~ of G . There is exactly one arc (i; j ) in H such that the sum xi + xj lies in the same coset modulo T as the di erence w ? v. Put t = (w ? v)(xi + xj )? , which belongs ~ v contains the arc to T . Further, put y = txi + v and z = txj + v. Evidently, G (y; z). On the other hand, y = ?txj + w and z = ?txi + w hold. It is noteworthy that the eld element ?1 lies in T since jT j is even. So with t also ?t is in T . Hence ~ w . We obtain that the union of any two pages the arc (z; y) belongs to the page G in G contains a digon. But there are exactly q(q ? 1)=2 di erent pairs of pages in G , and D~ q contains just q(q ? 1)=2 digons. It turns out that every pair of distinct pages has precisely one digon in its union. This concludes the proof. 1
We continue our study with odd primes q where (q ? 1)=e(H~ ) happens to be odd. Unfortunately, this case needs a slightly stronger argument. At this point it is convenient to introduce some basic facts on decompositions of graphs and digraphs into stars which will be used later on. For any positive integer m, let Sm denote the (undirected) m-star, i.e. the complete bipartite graph K ;m with m edges and m + 1 vertices. For m 2, it is customary to call the vertex of degree m in Sm the centre of the star. Lemma 10. Every simple graph H admits a decomposition into copies of 2S and at most one star, or a decomposition into copies of 2S and a triangle. +1
1
+1
2
2
Proof. We use induction on the number of edges in H. If H has only one edge or less, than the claim is trivially true. Suppose, the statement holds for all simple graphs with at most b ? 1 edges. We have to verify that it remains true for every simple graph H with b edges. First assume, H contains two vertex-disjoint edges. Together, they form a copy of 2S . Let H0 be the graph obtained from H by deleting these two edges. By the induction hypothesis, H0 admits an appropriate decomposition, and so H does. Conversely assume, any two edges in H share a vertex. Then H is star or a triangle. But then the claim trivially holds. 2
From Sm we obtain a directed m-star by assigning a direction to every edge in Sm . By S~ dm we shall denote the directed m-star whose centre has indegree d and, consequently, outdegree m ? d. When m = 1, there exists (up to isomorphism) ~ . only one directed 1-star which will be denoted by S The argument of Lemma 10 applies also to decompositions of digraphs. Here, we shall only state a slightly re ned result for digraphs with an even number of arcs. +1
+1
+1
2
14
r
r
r
r
r
r
r
*r
r
r
r
r
r
~ ,S ~ ,S ~ and S ~ . Figure 1: The digraphs 2S 2
0 3
1 3
2 3
Lemma 11. Every simple digraph H~ with an even number of arcs admits a decom~ , of S ~ , of S ~ and at most one copy of S ~ , such that all position into copies of 2S ~ ,S ~ and S ~ in this decomposition have the same centre. copies of S 2
0 3
1 3
2 3
0 3
2 3
1 3
~ admits a decomposition into copies of 2S ~ 2 and Proof. By Lemma 10 the digraph H ~ dm+1 with centre z. Note at most one directed star. Suppose, this star is a copy of S that m is even by assumption. ~ dm+1 admits a decomposition into d=2 copies of S ~ 23 and (m ? d)=2 If d is even too, S ~ 03. If d is odd, then S ~ dm+1 admits a decomposition into (d ? 1)=2 copies of copies of S S~ 23, (m ? d ? 1)=2 copies of S~ 03 and one copy of S~ 13. This concludes the proof. We are now in the position to establish a result similar to Lemma 9 for odd prime powers q where (q ? 1)=e(H~ ) is odd. Lemma 12. Let H~ be a simple digraph with non-negative cyclomatic number. There ~ ) such that every odd prime power q q with q exists a constant q = q (H e(H~ ) + 1 mod 2e(H~ ) belongs to IH~ . 12
12
12
~ ) is odd, it turns out that there is no odd prime power q satisfying Proof. When e(H the necessary condiditon q ? 1 e(H~ ) mod 2e(H~ ). So henceforth, we suppose that e(H~ ) is even. Put k = v(H~ ) and b = e(H~ ). Again, we consider V = f0; : : : ; q ? 1g to be the nite eld GF (q) of order q = ab + 1. By Lemma 11, the digraph H~ is decomposable ~ 1 or a directed into subdigraphs ~F1; : : : ; ~Fb=2 such that each of them is a copy of 2S 2-star. We denote the arcs in subdigraph ~Fl (l = 1; : : : ; b=2) by al and ab=2+l. The collection B = fa1; : : : ; abg of all these arcs forms the arc set of H~ . Let W = f1; : : : ; kg be the vertex set of H~ . Without loss of generality, let 1 denote the common centre of all directed 2-stars among ~F1; : : : ; ~Fb=2. If, in particular, the ~ 13, its arcs should be (2; 1) decomposition of H~ contains a digraph isomorphic to S and (1; 3). 15
Moreover, if either (2; 3) or (3; 2) occurs as an arc in H~ , then we may assume that this arc belongs to the subdigraph ~Fb= . In other words, suppose ab= = (3; 2) or ab = (2; 3) or neither of these pairs is an arc in H~ . As usual, let ? be the multiplicative group of GF (q) with the unique subgroup T of cardinality a and index b. When w denotes a generator of ?, the cosets modulo T are the sets Ti = wiT . In order to apply Lemma 5 we de ne a map C : W ! ?=T by 8 Tj? if g = h = i = j 2 W; > > > l Tb= l if g = h = i < j and (j; i) = al 2 B; > > : T otherwise: 2
2
4
1
2+
0
Put q (H~ ) = q (b; k). For prime powers q q (H~ ), Lemma 6 ensures the existence of a k-tuple x satisfying (8). ~ v in the decomposition G of the complete digraph D ~ q. Next we construct the pages G Fix v 2 V and t 2 T . The digraph obtained from H~ by assigning the vertex txj + v to each vertex j in H~ will be denoted by H~ v (t). Similarly to Lemma 9, the digraphs H~ v (t), t 2 T , are pairwise vertex-disjoint for xed v 2 V , and do not contain v as a vertex. Thus, the union G~ v of the digraphs H~ v (t), t 2 T , forms an idempotent H~ -family aH~ . As in Lemma 12 the collection G = fG~ v : v 2 V g turns out to be a decomposition ~ q . This is due to the observation that the di erences of the complete digraph D xj ? xi taken over all the arcs (i; j ) in H~ form again a system of representatives of the cosets modulo T . We now consider the question whether G is orthogonal, too. Let G~ v and G~ w be two distinct pages of G . Let Tm be the coset modulo T containing the di erence w ? v. Since ?1 lies in the coset Tb= this implies v ? w 2 Tb= m. Thus, we may suppose that 0 m < b=2. Consider the subdigraph F~ b= ?m of H~ with arcs ab= ?m = (i; j ) and ab?m = (i0; j 0). Due to our choice of the map C , we have xj ? xi 2 Tb= ?m and xj0 ? xi0 2 Tb?m. Put t = (w ? v)(xj ? xi)(xj xj0 ? xixi0 )? ; 12
6
12
2
2+
2
2
2
1
and
t0 = (w ? v)(xj0 ? xi0 )(xj xj0 ? xixi0 )? : These settings are legal as long as xj xj0 di ers from xixi0 . This diculty is overcome when the di erence xj xj0 ? xixi0 lies in coset T or Tb= modulo T . 1
0
16
2
If xj xj0 ? xixi0 2 T , we obtain t 2 TmTb= ?mT = Tb= and t0 2 TmTb?m T = T . Thus, the elements ?t and t0 lie in the subgroup T of ?. We put y = t0xi + v = (?t)xj0 + w and z = t0xj + v = (?t)xi0 + w. Hence the arc (y; z) lies in H~ v (t0) and, consequently, in the page G~ v of G . For the same reason the reverse arc (z; y) ~ w. belongs to H~ w (?t) and therefore to the page G Conversely, if xj xj0 ? xixi0 2 Tb= , this implies t 2 TmTb= ?mTb= = T and t0 2 TmTb?mTb= = Tb= . This time, t and ?t0 belong to the subgroup T . We put y = txi0 + v = (?t0)xj + w and z = txj0 + v = (?t0)xi + w. Again we nd (y; z) ~ v as well as (z; y) in the page G ~ w . This proves that any two distinct in the page G pages of G have at least one digon in their union. As mentioned in the proof of Lemma 9 this suces to verify that G is orthogonal. It remains to check that xj xj0 ? xixi0 actually is in T or Tb= . Four cases arise. First, ~ . Then the vertices i; i0; j; j 0 are assume the subdigraph ~Fb= ?m is isomorphic to 2S pairwise distinct. Due to our choice of the map C , this implies xj xj0 ? xixi0 2 T or xixi0 ? xj xj0 2 T depending on whether maxfi; i0; j; j 0g is in fj; j 0g or not. However, the latter relation may be rewritten as xj xj0 ? xixi0 2 Tb= . ~ . We immediately obtain i = i0 = 1 Secondly, assume ~Fb= ?m is isomorphic to S and j 6= j 0. This gives us xj xj0 ? xixi0 = xj xj0 ? x x 2 T . ~ . In this case we nd j = j 0 = 1 and i 6= i0. Next, let ~Fb= ?m be isomorphic to S This implies xixi0 ? x x 2 T , and evidentely, xj xj0 ? xixi0 2 Tb= . ~ . Thus we have to consider the subFinally, assume ~Fb= ?m is isomorphic to S digraph containing the arcs (2; 1) and (1; 3). Inspection gives us xj xj0 ? xixi0 = x (x ?x ). However both, x and x ?x lie in T . So we nd again xj xj0 ?xixi0 2 T . This completes the discussion of possible cases, and concludes the proof. 0
2
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2
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2
2
2
2
0
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2
0
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Finally, we record a similar result for even prime powers. However, this time we must not distinguish between several classes of even prime powers. Lemma 13. Let H~ be a simple digraph with non-negative cyclomatic number. There ~ ) such that every even prime power q q with exists a constant q = q (H ~ q 1 mod e(H) lies in IH~ . 13
13
13
Proof. The argument of Lemma 9 can easily be transfered to the case of even prime powers. In particular, we have to use Lemma 5 instead of Lemma 4 to ensure the existence of an appropriate k-tuple x. Furthermore, the essential observation that ?1 lies in the subgroup T is trivial for even q: in elds of even order we always have ?1 = 1.
Summing up the results from this section, we obtain the following observation. 17
Corollary 14. For every simple digraph H~ with non-negative cyclomatic number the set IH~ is non-empty and di erent from f1g. ~ ) be the supremum of the constants q9, q12 and q13. By virtue Proof. Let q14 = q14(H of Dirichlet's theorem there exists a prime p larger than q14 such that q 1 mod e(H~ ) holds. However, exactly one of the existence lemmas stated in this section applies to q. Thus IH~ contains q and satis es the claim.
4 PBD{closure The above discussion ensures the existence of prime powers in the set IH~ of admissible values. In order to study the set IH~ in greater detail we require the concept of pairwise balanced designs. Consider a positive integer n and a subset K of the positive integers. A pairwise balanced design PBD(n; K ) is a pair (V; F ) consisting of an n-set V and a collection F of subsets F V (called blocks ) such that every pair of distinct elements lies in exactly one block of F and all block sizes belong to K . The PBD-closure of a given set K of positive integers consists of all integers n admitting a pairwise balanced design PBD(n; K ). Consequently, the set K is said to be PBD-closed i K equals its PBD-closure. For a rigorous treatment of pairwise balanced designs and related topics, the reader should consult e.g. [3]. Examining the set IH~ for a given digraph H~ , we obtain the following result. Lemma 15. For every simple digraph H~ , the set IH~ is PBD-closed.
~ has non-negative cyclomatic Proof. According to Lemma 1, we may assume that H number. Let n be an integer from the PBD-closure of IH~ . Thus we nd a pairwise balanced design (V; F ) of order n and block sizes from IH~ . For every block ~ FxjF j g be an idempotent orthogonal F = fx1; : : : ; xjF jg 2 F , let G F = fG~ Fx1 ; : : : ; G ~ jF j by an H~ -family G~ F of size (jF j ? 1)=e(H~ ). decomposition of D For brevity, the set of all blocks F 2 F containing a xed element x is said to be ~ x be the union of all the digraphs the ower on x in F and denoted by F (x). Let G G~ Fx with F 2 F (x). Obviously, G~ x is again idempotent and an H~ -family of size ~ x) = n ? 1 arcs. (n ? 1)=e(H~ ). Thus, G~ x has just e(G Due to the de nition of pairwise balanced designs, every pair x,y of distinct elements from V belongs to exactly one block F 2 F . In G F there exists precisely one digraph 18
G~ Fz containing the arc (x; y). Hence, G~ z is the only page in G containing (x; y) and, ~ n. consequently, G is a decomposition of D ~ x and G~ y be two distinct digraphs in G . There is exactly one Furthermore, let G block F in F containing both, x and y. Since G F is an orthogonal decomposition, the union of G~ Fx and G~ Fy contains a digon. Due to our construction, this digon lies ~ x and G~ y , too. This shows that the union of any two distinct pages in the union of G in G contains at least (and thus exactly) one digon. A set K of positive integers is said to be eventually periodic with period i , with some integer r 2 K all suciently large integers n r mod lie in K , too. The notion of eventual periodic sets was introduced by Wilson [17] who proved the following criterion which is among the most interesting general theorems in design theory. Theorem 16 (Wilson). Let K be a non-empty PBD-closed set di erent from f1g. Then K is eventually periodic with period (K ) = g:c:dfn(n ? 1) : n 2 K g. In addition, K has period = (K )=2 i is odd and 0 mod (K ) holds, where (K ) = g:c:d:fn ? 1 : n 2 K g. On applying Wilson's theorem to the set IH~ we are able to verify the following property. Lemma 17. For every simple digraph H~ with non-negative cyclomatic number, the ~ ). set IH~ is eventually periodic with period e(H Proof. The eventual periodicity of IH~ follows immediately from Lemma 15 and Wilson's theorem. It remains to determine a period of IH~ . By Lemma 1, we have n ? 1 0 mod e(H~ ) for every n 2 IH~ . Hence, (IH~ ) is divisible by e(H~ ). Suppose, e(H~ ) is even. By Lemma 8 we nd two primes p; q q14 with p q 1 mod e(H~ )=2 and g:c:d:fp(p ? 1); q(q ? 1)g = e(H~ ). Both, p and q are odd and relatively prime to e(H~ )=2. Therefore, e(H~ ) divides p ? 1 as well as q ? 1. It turns out that p; q 2 IH~ and (IH~ ) = e(H~ ) holds. We now come to the case in which e(H~ ) is odd. This time, (IH~ ) is divisible by 2e(H~ ). Again by Lemma 7, there are primes p; q 2 IH~ with g:c:d:fp(p ? 1); q(q ? 1)g = 2e(H~ ) such that (IH~ ) = 2e(H~ ). Moreover, (IH~ ) is divisible by e(H~ ), too. Hence (IH~ ) is either e(H~ ) or 2e(H~ ) depending on the existence of even integers in IH~ . In order to nd even integers in IH~ , consider the Euler totient function (n) providing the number of positive integers m n with g:c:d:fm; ng = 1. Since e(H~ ) is odd, we obtain 2 (e(H~ )) 1 mod e(H~ )
19
due to the Euler-Fermat Theorem (see e.g. [11]). By Lemma 13 there is a positive integer with 2 e H~ 2 IH~ . Thus we obtain (IH~ ) = e(H~ ). Finally, Wilson's theorem settles our claim. ( ( ))
We are now in the position to establish our main result that the set IH~ contains almost all integers n satisfying the necessary conditions given by Lemma 1. We record this as follows: Theorem 18. Let H~ be a simple digraph with non-negative cyclomatic number. ~ ) such that for every n n with n 1 mod There exists a constant n = n (H ~ n by an H~ -family. e(H~ ) there is an idempotent orthogonal decomposition of D 18
18
18
5 Conclusions Let H~ be a simple digraph with v(H~ ) vertices and e(H~ ) arcs. In the preceding sections we proved the existence of an orthogonal decomposition by an H~ -family for all suciently large complete digraphs whenever e(H~ ) v(H~ ) holds. On the other hand, it is easy to check similar to Lemma 2 that there is no orthogonal decomposition by H~ -families if e(H~ ) v(H~ ) ? 2. The remaining case, namely e(H~ ) = v(H~ ) ? 1 is of special interest. Here an orthogonal decomposition by an H~ -family will only exist if this family is the digraph H~ itself. As pointed out in Section 1, this problem is unsolved in general, and the methods used in this article do not apply to this class of digraphs. Fortunately, the portion of simple digraphs with cyclomatic number -1 is considerably small. Theorem 19. For almost every simple digraph H~ there exists a constant m = m (H~ ) such that for every m m there is an orthogonal decomposition by mH~ . 19
19
19
~ has non-negative cyclomatic number. Choose m19 such that Proof. Suppose H m19e(H~ ) + 1 n18 holds. By Theorem 18 there exists an orthogonal decomposition ~ n by mH~ if n = me(H~ ) + 1 n18, i.e. m m19 holds. of D It remains to show that almost all simple digraphs have non-negative cyclomatic number. Let u(n; b) and l(n; b) denote the numbers of unlabeled and labeled simple digraphs with n vertices and b arcs, respectively. Moreover, let u(n) and l(n) be the total numbers of unlabeled and labeled simple digraphs with n vertices. It is easily veri ed, that l(n; b) =
?n
b b 2 2
20
and
l(n) =
1 X b=0
n l(n; b) = 3(2)
hold. In addition, it is well-known that almost every simple digraph has trivial automorphism group, i.e. the proportion n!u(n)=l(n) tends to 1 for n ! 1 (cf. [4]). Finally, let u?(n) and l?(n) denote the numbers of unlabeled and labeled simple digraphs with n vertices and negative cyclomatic number, respectively. We have
l?(n) =
n?1 X b=0
l(n; b) =
n?1 ?n X 2
b=0
b
2b
n?1 b X n b=0
bnn=3n 2 2 2
2 log3
n:
This implies ? ? ? n 0 uu((nn)) lu((nn)) = n!ll(n()n) nl!u(n(n) ) 3 n 3 n?( 2) nl!u(n(n) ) = o(1) n!lu(n(n) ) and consequently u?(n)=u(n) tends to 0 for n ! 1. This concludes the proof. 3 log
The statement of Theorem 19 is remarkable since it is widely believed that there exists an orthogonal decomposition of a complete digraph by almost all simple digraphs. Hence, the constant m (H~ ) is expected to be one for almost all H~ .However, a positive answer to this conjecture requires a proof which does not use Wilson's theorem on eventually periodic sets. 19
References [1] B. Alspach, K. Heinrich, and G. Liu. Orthogonal factorizations of graphs. In J.H. Dinitz and D.R. Stinson, editors, Contemporary Design Theory, chapter 2, pages 13{40. J. Wiley, New York, 1992. [2] F.E. Bennett and L. Wu. On minimum matrix representation of closure operations. Discr. Appl. Math., 26:25{40, 1990. [3] T. Beth, D. Jungnickel, and H. Lenz. Design Theory. BI, Mannheim, 1985. [4] P.J. Cameron. Automorphism groups of graphs. In L.W. Beineke and R.J. Wilson, editors, Selected topics in graph theory, volume 2, pages 89{127. Academic Press, London, 1983. 21
[5] J. Demetrovics, Z. Furedi, and G.O.H. Katona. Minimum matrix representations of closure operations. Discr. Appl. Math., 11:115{128, 1985. [6] J. Demetrovics and G.O.H. Katona. Extremal combinatorial problems in relational database. In Fundamentals of computation theory, volume 117 of Lecture Notes in Computer Science, pages 110{119. Springer, Berlin, 1981. [7] B. Ganter and H.-D.O.F. Gronau. On two conjectures of Demetrovics, Furedi and Katona on partitions. Discrete Math., 88:149{155, 1991. [8] A. Granville, H.-D.O.F. Gronau, and R.C. Mullin. On a problem of hering concerning orthogonal double covers of Kn . J. Comb. Theory, A(72):345{350, 1995. [9] F. Hering. Balanced pairs. Ann. Discrete Math., 20:177{182, 1984. [10] F. Hering and M. Rosenfeld. Problem number 38. In K. Heinrich, editor, Unsolved Problems: Summer Research Workshop in Algebraic Combinatorics. Simon Fraser University, 1979. [11] A.N. Parshin and I.R. Shafarevich. Number Theory. Springer, Berlin, 1990. [12] A. Rausche. On the existence of special block designs. Rostock. Math. Kolloq., 35:13{20, 1988. [13] R.Lidl and H.Niederreiter. Finite elds. Cambridge University Press, Cambridge, 1987. [14] W.M. Schmidt. Equations over nite elds. Springer, Berlin, 1976. [15] A. Weil. On some exponential sums. Proc. Nat. Acad. Sci. USA, 34:204{207, 1948. [16] R.M. Wilson. Cyclotomy and di erence families in elementary abelian groups. J. Number Theory, 4:17{42, 1972. [17] R.M. Wilson. An existence theory for pairwise balanced designs II. J. Comb. Theory (A), 13:246{270, 1972. [18] R.M. Wilson. Construction and uses of pairwise balanced designs. In M. Hall Jr. and J.H. van Lint, editors, Combinatorics, volume 55 of Math. Centre Tracts, pages 18{41. Mathematisch Centrum, Amsterdam, 1974. [19] R.M. Wilson. An existence theory for pairwise balanced designs III. J. Comb. Theory (A), 18:71{79, 1975. 22
