Orthogonal Decompositions of Complete Digraphs

Orthogonal Decompositions of Complete Digraphs

Sven Hartmann

FB Mathematik, Universitat Rostock, 18051 Rostock, Germanye-mail: [email protected]

Abstract. A family G of isomorphic copies of a given digraph ~GG is said to be an orthogonaldecomposition of the complete digraph ~DDn by ~GG, if every arc of ~DDn belongs to exactly onemember of G and the union of any two different elements from G contains precisely onepair of reverse arcs.

Given a digraph ~HH, an ~HH-family m~HH is the vertex-disjoint union of m copies of ~HH. In thispaper, we consider orthogonal decompositions by ~HH-families. Our objective is to prove theexistence of such an orthogonal decomposition whenever certain necessary conditions holdand m is sufficiently large.

1. Introduction

A collection G of subdigraphs of a digraph ~DD is said to be a decomposition of ~DD iffevery arc of ~DD belongs to exactly one member of G. Throughout, the subdigraphsin G are called the pages of the decomposition. If, in addition, all pages in G areisomorphic to a given digraph ~GG, we speak of a decomposition by ~GG.

For any positive integer n, let ~DDn denote the complete (symmetric) digraph onn vertices. A decomposition G of ~DDn is said to be orthogonal if the union of anytwo distinct pages in G contains exactly one digon, i.e. a copy of ~DD2.

Decomposition problems of graphs have received much attention during thelast two decades. Orthogonal decompositions were studied first by Hering andRosenfeld [10], who asked for an orthogonal decomposition of ~DDn by a directedðn� 1Þ-cycle. Even though this problem is far from being solved, it gave rise to aconsiderable number of results concerning orthogonal decompositions by specialdigraphs. 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 seeAlspach, Heinrich and Liu [1].

For a digraph ~HH, let m~HH denote the vertex-disjoint union of m copies of ~HH andcall it an ~HH-family of size m. In this paper, we concentrate on orthogonal de-compositions by ~HH-families. In the sequel, let ~HH always denote a simple digraph,i.e. without multiple arcs, loops and digons.

Let the vertex set of ~DDn be V ¼ f0; 1; . . . ; n� 1g. A page ~GGv is said to be idem-potent iff it contains vertex v as a singleton (or does not contain vertex v at all). Anorthogonal decomposition shall be called idempotent iff all its pages are idempotent.

Graphs and Combinatorics (2002) 18:285–302

Graphs andCombinatorics� Springer-Verlag 2002

When considering keys and antikeys in databases, Demetrovics and Katona [6]asked for idempotent orthogonal decompositions of ~DDn by directed 3-cyclefamilies. A first, partial answer to this question was given by Demetrovics, Furediand Katona [5]. Rausche [12] proved that there is no such decomposition of ~DD10.Finally, the problem was completely solved by Bennett and Wu [2] and, inde-pendently, by Ganter and Gronau [7] who proved the existence of an idempotentorthogonal decomposition of ~DDn by a directed-3cycle family iff n � 1 mod 3 andn 6¼ 10.

More recently, Granville, Gronau and Mullin [8] proved that for every k 3there exists an idempotent orthogonal decomposition of ~DDn by a directed k-cyclefamily whenever the necessary condition n � 1 mod k holds and n is sufficientlylarge. This question was asked in the undirected case by Hering [9].

Our objective is to generalize the latter result to arbitrary digraph families. Weshall prove the existence of an idempotent orthogonal decomposition of ~DDn by an~HH-family for almost all n whenever certain necessary conditions hold. Thesepreconditions are given by the following lemma.

Lemma 1. Let ~HH be a simple digraph with v(~HH) vertices and e(~HH) arcs. If thereexists an idempotent orthogonal decomposition of ~DDn by an ~HH-family thenv(~HHÞ e(~HH) and n � 1 mod e(~HH) hold.

Proof. Assume, there exists an idempotent orthogonal decomposition of ~DDn by an~HH-family ~GG ¼ m~HH of size m. For the numbers of vertices and arcs in ~DDn we obtain

vð~DDnÞ ¼ n vð~GGÞ þ 1 ¼ m vð~HHÞ þ 1

and

eð~DDnÞ ¼ nðn� 1Þ ¼ n eð~GGÞ ¼ nm eð~HHÞ;

respectively. This implies eð~HHÞ vð~HHÞ as well as n� 1 � 0 mod eð~HHÞ. h

The difference eð~HHÞ � vð~HHÞ is usually called the cyclomatic number of the di-graph ~HH. Thus we are looking for orthogonal decompositions by ~HH-families where~HH has non-negative cyclomatic number.

2. Preliminaries

In this section we shall assemble some terminology and basic results which will berequired in the sequel. Throughout, let q denote a prime power, and let a and b bepositive integers such that q ¼ abþ 1 holds. Consider V ¼ f0; . . . ; q� 1g to be thefinite field GF ðqÞ of order q. Let C denote its cyclic multiplicative group con-taining all non-zero elements of GF ðqÞ, and let T be its unique subgroup ofcardinality a and index b. If w denotes a generator of C, then is clearly generatedby wb.

286 S. Hartmann

A character on C is a map v from C to the complex numbers satisfyingjvðxÞj ¼ 1 and vðxyÞ ¼ vðxÞvðyÞ for all x; y 2 C. The characters on C form again acyclic group of order q� 1, the dual group C� of C.

It will be convenient to extend the usual definition of a character v on C byputting

vð0Þ ¼ 1 if v ¼ v0,0 otherwise,

�

where v0 is the principal character with vðxÞ ¼ 1 for all x 2 C.Let T � be the unique subgroup of index a and cardinality b in C�. If w denotes

a generator of C� then T � is clearly generated by wa. The interplay between thecharacters in T � and the elements in T is given by the following well-knownequality (see e.g. [14]):

Xv2T �

vðxÞ ¼Xb�1

j¼0

wajðxÞ ¼b if x 2 T ,1 if x ¼ 0,0 otherwise.

(ð1Þ

Let f be a polynomial of positive degree d over GF ðqÞ, and let E be an extensionfield of GF ðqÞ. Then f is said to split in E iff f can be written as a product of linearfactors over E, i.e. iff there exist elements x1; . . . ; xd 2 E such that

f ðxÞ ¼ zðx� x1Þ � � � ðx� xdÞ;

where z is the leading coefficient of f . The values x1; . . . ; xd are called the roots off . Note that there is a (up to isomorphism) unique smallest extension field con-taining all roots of f , the splitting field of f . For more detailed information onpolynomials and splitting fields, we refer to [13].

We are interested in sums of the formP

x2GF ðqÞ vðf ðxÞÞ, where v is a fixedcharacter and f a fixed polynomial over GF ðqÞ. The problem of evaluatingcharacter sums is known to be difficult in general. One usually has to be satisfiedwith estimates for the absolute value of such sums. We shall make use of thefollowing 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 isnot that b-th power of a polynomial over GF(q). Let r be the number of pairwisedistinct roots of f in a splitting field of f. Then

jX

x2GF ðqÞwaðf ðxÞÞj ðr � 1Þq1=2

holds, where w denotes a generator of the dual group C�.If X and Y are two subsets of C, their product XY is defined as the set

fxy 2 C : x 2 X and y 2 Y g. If X ¼ fxg we shall write xY instead of fxgY , forbrevity. Moreover, let X�1 denote the set fx�1 : x 2 Xg of all inverses of theelements in X .

Orthogonal Decompositions of Complete Digraphs 287

Let w be a generator of C. We now consider the cosets Ti ¼ wiT of C modulo T ,where i is some integer. Ti and Tj denote the same coset whenever i � j mod bholds. The cosets themselves form again a group C=T known as the factor groupof C (with respect to T ). The order of C=T is just ðq� 1Þ=jT j ¼ b.

We continue with a somewhat technical result that forms the foundation ofwhat follows.

Lemma 3. Let b be a positive integer and r,s be non-negative integers. There is aconstant q3 ¼ q3ðb; r; sÞ such that for all prime powers q q3 with q � 1 mod b, forall r-sets U � GF ðqÞ, for all s-sets Z � GF ðqÞ and for all maps h : U ! C=T thereexists an element x 2 GF ðqÞ satisfying

x =2Z; ð2Þ

x� u 2 hðuÞ ð3Þ

for every u 2 U .

Proof. The claim is trivial when r ¼ 0 holds. Conversely, let r be positive. Anelement x 2 GF ðqÞ lies in coset Ti modulo T iff w�ix lies in T , where w denotes asabove a generator of C. For every integer i and for every element x 2 GF ðqÞ, put

SiðxÞ ¼1

b

Xv2T �

vðw�ixÞ ¼1 if x 2 Ti,1b if x ¼ 0,0 otherwise.

(ð4Þ

The idea is to use such sums to construct an expression which has value 1 whenever(3) holds and becomes sufficiently small otherwise. For the sake of simplicity, wedefine a function i : U ! f0; . . . ; b� 1g given by hðuÞ ¼ TiðuÞ for every u 2 U .Next, we put

P ðxÞ ¼Yu2U

SiðuÞðx� uÞ:

Some consequences of (4) may be noted immediately. We have P ðxÞ ¼ 1 iff (3)holds. Moreover, PðxÞ is positive iff x� u 2 hðuÞ [ f0g holds for every u 2 U . Thisimplies 0 < P ðxÞ < 1 iff we have x� u 2 hðuÞ [ f0g for every u 2 U , and x lies inU . Hence there are at most r elements x satisfying 0 < PðxÞ < 1. We emphasizethat in this case we actually have 0 < P ðxÞ 1=b:

Finally, we put

Q ¼X

x2GF ðqÞP ðxÞ:

Then Q > r=b implies the existence of an element x 2 GF ðqÞ with P ðxÞ ¼ 1, i.e. (3)holds. The existence of an appropriate x satisfying (2) and (3) will follow ifQ > r=bþ s.

We shall show that Q becomes larger than r=bþ s if q is sufficiently large. Tobegin with, we find that

288 S. Hartmann

Q ¼ 1

brX

x2GF ðqÞ

Yu2U

Xv2T �

vðw�iðuÞðx� uÞÞ

¼ 1

brX

x2GF ðqÞ

Yu2U

Xb�1

j¼0

wajðw�iðuÞðx� uÞÞ;

where w denotes as above a generator of C�.Let M denote the set of all r-tuples j ¼ ðju : u 2 UÞ 2 f0; . . . ; b� 1gr. On

expanding and recombining Q, we obtain

Q ¼ 1

brXj2M

Xx2GF ðqÞ

Yu2U

wajuðw�iðuÞðx� uÞÞ

¼ 1

brXj2M

Xx2GF ðqÞ

Yu2U

waðw�juiðuÞðx� uÞjuÞ

¼ 1

brXj2M

Xx2GF ðqÞ

waðw�P

u2U juiðuÞYu2U

ðx� uÞjuÞ:

For every tuple j 2 M , put

zj ¼ w�P

u2U juiðuÞ;

and let

fjðxÞ ¼Yu2U

ðx� uÞju

be a polynomial of degreeP

u2U ju over GF ðqÞ:If j is the tuple with all zero-entries, simply denoted by o, then zo ¼ w0 ¼ 1 and

fo is the constant polynomial of value 1 for all x 2 GF ðqÞ. Hence, we obtainXx2GF ðqÞ

waðzofoðxÞÞ ¼X

x2GF ðqÞwað1Þ ¼ q: ð5Þ

Conversely, let j be different from o. The roots of fj are just the elements u 2 Uwhich satisfy ju > 0. Hence, fj has at most r distinct roots. Since all compo-nents ju in j are smaller than b, and at least one of them is positive, fj is notthe b-th power of a polynomial over GF ðqÞ. Applying Weil’s theorem, weobtain

jX

x2GF ðqÞwaðzjfjðxÞÞj ¼ jwaðzjÞjj

Xx2GF ðqÞ

waðfjðxÞÞj ðr � 1Þq1=2 ð6Þ

for every j 6¼ o:


Combining (5) and (6) we obtain

Q� 1

brX

x2GF ðqÞwaðzofoðxÞÞ ¼

1

brXj6¼o

Xx2GF ðqÞ

waðzjfjðxÞÞ

jQ� 1

brX

x2GF ðqÞwaðzofoðxÞÞj

1

brXj6¼o

jX

x2GF ðqÞwaðzjfjðxÞÞj

jQ� 1

brqj 1

brðbr � 1Þðr � 1Þq1=2

jQ� 1

brqj ðr � 1Þq1=2

and thus

Q 1

brq� ðr � 1Þq1=2;

which tends to infinity when q ! 1.So x can be chosen satisfying (2) and (3) if q is large enough. This concludes the

proof. A straightforward, but tedious calculation shows that q3 ¼ q3ðb; r; sÞ ¼b2rðr þ sÞ2 þ 1 will be a lower bound for q to ensure the existence of an appro-priate x 2 GF ðqÞ.

In the sequel, let Xk denote the set f1; . . . ; kg. A choice over GF ðqÞ is a mapC : X2

k ! C=T assigning a coset modulo T to each pair ði; jÞ of integers i; j 2 Xk.With the help of our Lemma 3 it is easy to verify that there exists a k-tuplex ¼ ðx1; . . . ; xkÞ of elements in C satisfying xj � xi 2 Cði; jÞ for every pair i; j 2 Xk

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 balanceddesigns [19]. For details, we refer to [3]. Our objective is to prove several gener-alizations of Wilson’s result.

Lemma 4. Let b, k be positive integers. There is a constant q ¼ q4ðb; kÞ such that forall odd prime powers q q4 with q � 1 mod b and for all maps C : X2

k ! C=T thereexists a k-tuple x of elements from GF ðqÞ satisfying

� xi þ xj 2 Cði; jÞ if i < j;

xj 2 Cði; jÞ if i ¼ j;

xi þ xj 2 Cði; jÞ if i > j

ð7Þ

for all pairs ði; jÞ 2 X2k .

Proof. We shall proceed by induction on k. For k ¼ 1 the claim is trivially true:just choose x1 from the coset Cð1; 1Þ which is non-empty for q 2.

Assume, the statement is true for k � 1, and we shall prove it for k, now. Letx ¼ ðx1; . . . ; xk�1Þ be a ðk � 1Þ-tuple satisfying (7) for each pair ði; jÞ 2 X2

k�1.

290 S. Hartmann

It remains to find an element xk 2 Cðk; kÞ with xk � xi 2 Cði; kÞ andxk þ xi 2 Cðk; iÞ for every i 2 Xk�1. It turns out that Lemma 3 ensures the existenceof such an element xk when putting Z ¼ ; andU ¼ f0; x1; . . . ; xk�1;�x1; . . . ;�xk�1g.Note that the elements 0, xi ði 2 Xk�1Þ and �xi ði 2 Xk�1Þ are pairwise distinct byvirtue of the induction hypothesis and since q is odd.

We must now define a map h on U by

hðuÞ ¼Cðk; kÞ if u ¼ 0,Cði; kÞ if u ¼ xi,Cðk; iÞ if u ¼ �xi

8<:

for every element u 2 U . Hence, we are looking for an element xk 2 GF ðqÞ suchthat xk � u 2 hðuÞ holds for every u 2 U . By Lemma 3, such an element existswhenever q q3ðb; 2k � 1; 0Þ: h

Unfortunately, for even prime powers, the statement of Lemma 3 is usuallyfalse since y � x ¼ y þ x holds for all elements x; y 2 GF ðqÞ if q is even. However,the following weaker result holds.

Lemma 5. Let b, k be positive integers. There is a constant q5 ¼ q5ðb; kÞ such thatfor all even prime powers q q5 with q � 1 mod b and for all maps C : X2

k ! C=Tthere exists a k-tuple x of elements from GF ðqÞ satisfying

xi þ xj ¼ �xi þ xj 2 Cði; jÞ if i < j;

xj 2 Cði; jÞ if i ¼ j

for all pairs ði; jÞ 2 X2k .

Proof. The proof works analogously to the proof of Lemma 4. This time weobtain q5 ¼ q3ðb; k; 0Þ. h

We now consider choices C over GF ðqÞ assigning a coset modulo T to every4-tuple ðg; h; i; jÞ 2 X4

k . Applying Lemma 3 we shall verify the following claimwhich holds for both, even and odd prime powers.

Lemma 6. Let b, k be positive integers. There is a constant q6 ¼ q6ðb; kÞ such thatfor all prime powers q q6 with q � 1 mod b and for all maps C : X4

k ! C=T thereexists a k-tuple x with elements from GF ðqÞ satisfying

xj 2 Cðg; h; i; jÞ if g ¼ h ¼ i ¼ j;

�xi þ xj 2 Cðg; h; i; jÞ if g ¼ h ¼ i < j;

�xgxh þ xixj 2 Cðg; h; i; jÞ if g h < i < j

or g < i < h < j or i < g h < j

ð8Þ

for all 4-tuples ðg; h; i; jÞ 2 X4k .

Proof. Again we shall proceed by induction on k. However, we shall prove theslightly stronger statement that there is a k-tuple x satisfying not only (8) but also


xgxhxi 6¼ xg0xh0xi0 ð9Þ

whenever two multisets fg; h; ig and fg0; h0; i0g over Xk are distinct.When k ¼ 1 the claim trivially holds: just choose x1 in the coset Cð1; 1; 1; 1Þ

which is non-empty for q 2. Assume, the statement is true for k � 1, and weshall prove it for k, now.

Let x ¼ ðx1; . . . ; xk�1Þ be a ðk � 1Þ-tuple satisfying (8) for each 4-tupleðg; h; i; jÞ 2 X4

k�1 as well as our additional property (9). We are interested in anelement xk 2 Cðk; k; k; kÞ with xk � xi 2 Cði; i; i; kÞ for every i 2 Xk�1 andxk � xgxhx�1

i 2 Cðg; h; i; kÞCði; i; i; iÞ�1 for every 3-tuple ðg; h; iÞ 2 Xþk�1, where

Xþk�1 ¼ fðg; h; iÞ 2 X3

k�1 : g h < i or g < i < h or i < g hg:

In order to apply Lemma 3, we put

U ¼ f0; x1; . . . ; xk�1g [ fxgxhx�1i : ðg; h; iÞ 2 Xþ

k�1g;

and

Z ¼ XXXX�1X�1 [ fz : z2 2 XXXX�1g [ fz : z3 2 XXXg;

where X denotes the set fx1; . . . ; xk�1g containing the components of the ðk � 1Þ-tuple x. On U we define a map h : U ! C=T by

hðuÞ ¼Cðk; k; k; kÞ if u ¼ 0,Cðj; j; j; kÞ if u ¼ xj,Cðg; h; i; kÞCði; i; i; iÞ�1 if u ¼ xgxhx�1

i

8<:

for every integer j 2 Xk�1 and every 3-tuple ðg; h; iÞ 2 Xþk�1:

In general, this definition will only be correct if the elements 0, xj ðj 2 Xk�1Þand xgxhx�1

i ððg; h; iÞ 2 Xþk�1Þ are pairwise distinct.

It is readily verified that the elements 0 and xj ðj 2 Xk�1Þ are pairwise distinctby the induction hypothesis. For the same reason, all the elements xgxhx�1

i aredifferent from 0.

Next suppose, there are two distinct 3-tuples ðg; h; iÞ and ðg0; h0; i0Þ in Xþk�1 such

that xgxhx�1i ¼ xg0xh0x�1

i0 holds. This implies xgxhxi0 ¼ xg0xh0xi and by virtue of (9) fork � 1, the multisets fg; h; i0g and fg0; h0; ig are equal. Since i 6¼ g; h, we obtaini ¼ i0. Moreover, g h and g0 h0 imply g ¼ g0 as well as h ¼ h0. Clearly, thisgives a contradiction.

Finally suppose, there is an integer j 2 Xk�1 and a 3-tuple ðg; h; iÞ 2 Xþk�1 such

that xj ¼ xgxhx�1i holds. This implies xixj ¼ xgxh, and again by the induction

hypothesis (9), the multisets fi; jg and fg; hg are equal. But i 6¼ g; h, which givesa contradiction.

By Lemma 3, there exists an element xk 2 GF ðqÞ satisfying xk � u 2 hðuÞ forevery u 2 U and xk=2Z if q q3ðb; k3; 6k5Þ. Note, that jU j k3 and jZj 6k5

hold.

292 S. Hartmann

To satisfy our additional property (9) we have to ensure xgxhxi 6¼ xg0xh0xi0whenever the multisets fg; h; ig and fg0; h0; i0g over Xk are distinct. If k is neither infg; 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 Xk

such that xgxhxi ¼ xg0xh0xi0 holds and k lies in fg; h; i; g0; h0; i0g. Thus we obtainxk 2 XXXX�1X�1 or x2

k 2 XXXX�1 or x3k 2 XXX . But such a relation contradicts

xk=2Z. This concludes the proof. h

The preceding lemmas require large prime powers q satisfying the additionalcondition q � 1 mod b for a given integer b. The existence of such prime powers isensured by Dirichlet’s well-known theorem on primes in arithmetic progressions(cf. [11]). We record this result for future reference.

Theorem 7 (Dirichlet). Let p7; b; r be positive integers with g:c:d:fb; rg ¼ 1. Thereexists a prime p p7 such that p � r mod b holds.

By a further investigation we may render the following slightly modifiedexistence result on primes in arithmetic progressions.

Lemma 8. Let p8; b; r be positive integers with g:c:d:f2b; rg ¼ 1. There exist twoprimes p; q with q > p p8 such that p � q � r mod b and g:c:d:fpðp � rÞ;qðq� rÞg ¼ 2b hold.

Proof. By Dirichlet’s theorem there is a prime p p8 with p � 4bþ r mod2bð2bþ rÞ, say p ¼ 2bð2bþ rÞk þ 4bþ r for some positive integer k. Putl ¼ ð2bþ rÞk þ 2, such that p ¼ 2bl þ r holds. Again by Dirichlet’s theoremthere is a prime q > p with q � 2bþ r mod 2blp, say q ¼ 2blpm þ 2bþ r for somepositive integer m. Clearly, p � q � r mod b holds. Moreover, we haveg:c:d:fpðp � rÞ, qðq� rÞg ¼ g:c:d:fpðp � rÞ; q� rg ¼ g:c:df2blp; 2blpm þ 2bg ¼2b g:c:d:flp; lpm þ 1g ¼ 2b. h

3. Main Constructions

For a given digraph ~HH, let I~HH denote the set of all integers n such that ~DDn

admits an idempotent orthogonal decomposition by an ~HH-family. Ourultimative aim in this section is to investigate the occurrence of prime powersin I~HH . In order to study orthogonal decompositions of complete digraphs~DDq where q is a prime power, we shall exploit the concepts presented inSection 2.

It is advantageous to distinguish several cases. To begin with, we considerodd prime powers q satisfying the additional property that ðq� 1Þ=eð~HHÞ iseven.

Lemma 9. Let ~HH be a simple digraph with non-negative cyclomatic number. Thereexists a constant q9 ¼ q9ð~HHÞ such that every odd prime power q q9 with q � 1mod 2eð~HHÞ lies in I~HH .


Proof. Put k ¼ vð~HHÞ and b ¼ eð~HHÞ. Consider V ¼ f0; . . . ; q� 1g to be the finitefield GF ðqÞ of order q ¼ abþ 1. Throughout, we shall use the terminology in-troduced in Section 2. In particular, let C be the multiplicative group of GF ðqÞ,and T its unique subgroup of cardinality a and index b. When w denotes agenerator of C, then the cosets modulo T are given by Ti ¼ wiT .

Let W ¼ f1; . . . ; kg and B ¼ fa1 : l ¼ 1; . . . ; bg be the vertex set and arc set of~HH, respectively. On W 2 we define a map C : W 2 ! C=T by

Cði; jÞ ¼Tj�1 if i ¼ j 2 W ,Tl if ði; jÞ ¼ al 2 B or ðj; iÞ ¼ al 2 B,T0 otherwise.

8<:

Put q9 ¼ q4ðb; kÞ and let q q9. Applying Lemma 4 we find a k-tuple x satisfyingxj 2 Tj�1 for all vertices j in ~HH, and xj � xi 2 Tl as well as xj þ xi 2 Tl for all arcsal ¼ ði; jÞ 2 B. It is worth mentioning that both the differences xj � xi and thesums xi þ xj taken over all arcs ði; jÞ in ~HH form a system of representatives of thecosets modulo T .

For every v 2 V and for every t 2 T , we construct a digraph ~HHvðtÞ as theisomorphic image of ~HH under the map j ! txj þ v for all vertices j in ~HH.

First, we shall show that for every fixed v 2 V the digraphs ~HHvðtÞ with t 2 T arepairwise vertex-disjoint. Assume, there are distinct elements t and t0 in T such thatthe digraphs ~HHvðtÞ and ~HHvðt0Þ share a vertex z. Then there are two vertices j and j0

in ~HH such that z ¼ txj þ v ¼ t0xj0 þ v holds. This implies txj ¼ t0xj0 . So xj and xj0 liein the same coset modulo T . Due to our choice of C, we obtain j ¼ j0 and con-sequently t ¼ t0, which gives a contradiction. Hence, for fixed v 2 V the digraphs~HHvðtÞ; t 2 T , form an ~HH-family a~HH which will be denoted by ~GGv. It should be

mentioned that ~GGv has exactly eð~GGvÞ ¼ aeð~HHÞ ¼ q� 1 arcs.

Secondly, we point out that ~GGv is idempotent for every v 2 V . Assume, v is avertex in ~GGv. This implies the existence of a vertex j in ~HH satisfying txj þ v ¼ v forsome element t 2 T . However, this contradicts xj 6¼ 0 for all j in ~HH. Thus ~GGv isidempotent.

Next, we shall check that the digraphs ~GGv; v 2 V , form a decomposition G of~DDq. Let G denote the collection of all digraphs ~GGv; v 2 V . Since each of the qdigraphs in G contains exactly q� 1 arcs, it suffices to show that every arc of ~DDq

belongs to at least one of the digraphs ~GGv.Let ðy; zÞ be an arbitrary arc in ~DDq. In ~HH we find exactly one arc ði; jÞ such that

the difference xj � xi lies in the same coset modulo T as the difference z� y. Putt ¼ ðz� yÞðxj � xiÞ�1, and v ¼ y � txi ¼ z� txj. Clearly, t is an element of T , and vlies in V . It is now easy to see that the digraph ~HHvðtÞ contains the arc ðy; zÞ, and sodoes the page ~GGv. We conclude that every arc of ~DDq lies in at least one page of G,and consequently, G forms a decomposition of the complete digraph ~DDq.

It still remains to prove that G is orthogonal. Let ~GGv and ~GGw be two distinctpages of G. There is exactly one arc ði; jÞ in ~HH such that the sum xi þ xj lies in thesame coset modulo T as the difference w� v. Put t ¼ ðw� vÞðxi þ xjÞ�1, whichbelongs to T . Further, put y ¼ txi þ v and z ¼ txj þ v. Evidently, ~GGv contains the

294 S. Hartmann

arc ðy; zÞ. On the other hand, y ¼ �txj þ w and z ¼ �txi þ w hold. It is noteworthythat the field element �1 lies in T since jT j is even. So with t also �t is in T . Hencethe arc ðz; yÞ belongs to the page ~GGw. We obtain that the union of any two pages inG contains a digon. But there are exactly qðq� 1Þ=2 different pairs of pages in G,and ~DDq contains just qðq� 1Þ=2 digons. It turns out that every pair of distinctpages has precisely one digon in its union. This concludes proof. h

We continue our study with odd primes q where ðq� 1Þ=eð~HHÞ happens to beodd. Unfortunately, this case needs a slightly stronger argument. At this point it isconvenient to introduce some basic facts on decompositions of graphs anddigraphs into stars which will be used later on.

For any positive integer m, let Smþ1 denote the (undirected) m-star, i.e. thecomplete bipartite graph K1;m with m edges and mþ 1 vertices. For m 2, it iscustomary to call the vertex of degree m in Smþ1 the centre of the star.

Lemma 10. Every simple graph H admits a decomposition into copies of 2S2 and atmost one star, or a decomposition into copies of 2S2 and a triangle.

Proof. We use induction on the number of edges in H. If H has only one edge orless, than the claim is trivially true. Suppose, the statement holds for all simplegraphs with at most b� 1 edges. We have to verify that it remains true for everysimple graph H with b edges.

First assume H contains two vertex-disjoint edges. Together, they form a copyof 2S2. Let H0 be the graph obtained from H by deleting these two edges. By theinduction 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 atriangle. But then the claim trivially holds. h

From Smþ1 we obtain a directed m-star by assigning a direction to every edgein Smþ1. By ~SSd

mþ1 we shall denote the directed m-star whose centre has indegree dand, consequently, outdegree m� d. When m ¼ 1, there exists (up to isomor-phism) only one directed 1-star which will be denoted by ~SS2.

The argument of Lemma 10 applies also to decompositions of digraphs. Here,we shall only state a slightly refined result for digraphs with an even number of arcs.

Lemma 11. Every simple digraph ~HH with an even number of arcs admits a decom-

position into copies of 2~SS2, of ~SS03, of ~SS2

3 and at most one copy of~SS1

3, such that all

copies of ~SS03, ~SS1

3 and ~SS23 in this decomposition have the same centre.

Proof. By Lemma 10 the digraph ~HH admits a decomposition into copies of 2~SS2

and at most one directed star. Suppose, this star is a copy of ~SSdmþ1 with centre z.

Note that m is even by assumption.

Fig. 1. The digraphs 2~SS2; ~SS03;~SS1

3 and ~SS23


If d is even too, ~SSdmþ1 admits a decomposition into d=2 copies of ~SS2

3 and

ðm� dÞ=2 copies of ~SS03. If d is odd, then ~SSd

mþ1 admits a decomposition into

ðd � 1Þ=2 copies of ~SS23; ðm� d � 1Þ=2 copies of ~SS0

3 and one copy of ~SS13. This

concludes the proof. h

We are now in the position to establish a result similar to Lemma 9 for oddprime powers q where ðq� 1Þ=eð~HHÞ is odd.

Lemma 12. Let ~HH be a simple digraph with non-negative cyclomatic number. Thereexists a constant q12 ¼ q12ð~HHÞ such that every odd prime power q q12 withq � eð~HHÞ þ 1 mod 2eð~HHÞ belongs to I~HH .

Proof. When eð~HHÞ is odd, it turns out that there is no odd prime power q satis-fying the necessary condition q� 1 � eð~HHÞ mod 2eð~HHÞ. So henceforth, we supposethat eð~HHÞ is even.

Put k ¼ vð~HHÞ and b ¼ eð~HHÞ. Again, we consider V ¼ f0; . . . ; q� 1g to be the

finite field GF ðqÞ of order q ¼ abþ 1. By Lemma 11, the digraph ~HH is decom-posable into subdigraphs ~FF1; . . . ;

~FFb=2 such that each of them is a copy of 2~SS2 or adirected 2-star.

We denote the arcs in subdigraph ~FFlðl ¼ 1; . . . ; b=2Þ by al and ab=2þl. Thecollection B ¼ fa1; . . . ; abg of all these arcs forms the arc set of ~HH.

Let W ¼ f1; . . . ; kg be the vertex set of ~HH. Without loss of generality, let 1denote the common centre of all directed 2-stars among ~FF1; . . . ;

~FFb=2. If, in par-ticular, the decomposition of ~HH contains a digraph isomorphic to ~SS1

3, its arcsshould be (2,1) and (1,3).

Moreover, if either (2,3) or (3,2) occurs as an arc in ~HH, then we may assumethat this arc belongs to the subdigraph ~FFb=2. In other words, suppose ab=2 ¼ ð3; 2Þor ab ¼ ð2; 3Þ or neither of these pairs is an arc in ~HH.

As usual, let C be the multiplicative group of GF ðqÞ with the unique subgroupT of cardinality a and index b. When w denotes a generator of C, the cosetsmodulo T are the sets Ti ¼ wiT . In order to apply Lemma 5 we define a mapC : W 4 ! C=T by

Cðg; h; i; jÞ ¼Tj�1 if g ¼ h ¼ i ¼ j 2 W ,Tl if g ¼ h ¼ i < j and ði; jÞ ¼ al 2 B,Tb=2þl if g ¼ h ¼ i < j and ðj; iÞ ¼ al 2 B,T0 otherwise.

8><>:

Put q12ð~HHÞ ¼ q6ðb; kÞ . For prime powers q q12ð~HHÞ, Lemma 6 ensures theexistence of a k-tuple x satisfying (8).

Next we construct the pages ~GGv in the decomposition G of the complete di-graph ~DDq. Fix v 2 V and t 2 T . The digraph obtained from ~HH by assigning thevertex txj þ v to each vertex j in ~HH will be denoted by ~HHvðtÞ. Similarly to Lemma 9,the digraphs ~HHvðtÞ; t 2 T , are pairwise vertex-disjoint for fixed v 2 V , and do notcontain v as a vertex. Thus, the union ~GGv of the digraphs ~HHvðtÞ; t 2 T , forms anidempotent ~HH-family a~HH.

296 S. Hartmann

As in Lemma 12 the collection G ¼ f~GGv : v 2 V g turns out to be a decom-position of the complte digraph ~DDq. This is due to the observation that thedifferences xj � xi taken over all the arcs ði; jÞ in ~HH form again a system ofrepresentatives of the cosets modulo T .

We now consider the question whether G is orthogonal, too. Let ~GGv and ~GGw betwo distinct pages of G. Let Tm be the coset modulo T containing the differencew� v. Since �1 lies in the coset Tb=2 this implies v� w 2 Tb=2þm. Thus, we maysuppose that 0 m < b=2. Consider the subdigraph ~FFb=2�m of ~HH with arcsab=2�m ¼ ði; jÞ and ab�m ¼ ði0; j0Þ. Due to our choice of the map C, we havexj � xi 2 Tb=2�m and xj0 � xi0 2 Tb�m. Put

t ¼ ðw� vÞðxj � xiÞðxjxj0 � xixi0 Þ�1;

and

t0 ¼ ðw� vÞðxj0 � xi0 Þðxjxj0 � xixi0 Þ�1:

These settings are legal as long as xjxj0 differs from xixi0 . This difficulty is overcomewhen the difference xjxj0 � xixi0 lies in coset T0 or Tb=2 modulo T .

If xjxj0 � xixi0 2 T0, we obtain t 2 TmTb=2�mT0 ¼ Tb=2 and t0 2 TmTb�mT0 ¼ T0.Thus, the elements �t and t0 lie in the subgroup T of C. We puty ¼ t0xi þ v ¼ ð�tÞxj0 þ w and z ¼ t0xj þ v ¼ ð�tÞxi0 þ w. Hence the arc ðy; zÞ lies in~HHvðt0Þ and, consequently, in the page ~GGv of G. For the same reason the reverse arcðz; yÞ belongs to ~HHwð�tÞ and therefore to the page ~GGw.

Conversely, if xjxj0 � xixi0 2 Tb=2, this implies t 2 TmTb=2�mTb=2 ¼ T0 andt0 2 TmTb�mTb=2 ¼ Tb=2. This time, t and �t0 belong to the subgroup T . We puty ¼ txi0 þ v ¼ ð�t0Þxj þ w and z ¼ txj0 þ v ¼ ð�t0Þxi þ w. Again we find ðy; zÞ in thepage ~GGv as well as ðz; yÞ in the page ~GGw. This proves that any two distinct pages ofG have at least one digon in their union. As mentioned in the proof of Lemma 9this suffices to verify that G is orthogonal.

It remains to check that xjxj0 � xixi0 actually is in T0 or Tb=2. Four cases arise.First, assume the subdigraph ~FFb=2�m is isomorphic to 2~SS2. Then the verticesi; i0; j; j0 are pairwise distinct. Due to our choice of the map C, this impliesxjxj0 � xixi0 2 T0 or xixi0 � xjxj0 2 T0 depending on whether maxfi; i0; j; j0g is infj; j0g or not. However, the latter relation may be rewritten as xjxj0 � xixi0 2 Tb=2.

Secondly, assume ~FFb=2�m be isomorphic to ~SS03. We immediately

obtain i ¼ i0 ¼ 1 and j 6¼ j0. This gives us xjxj0 � xixi0 ¼ xjxj0 � x1x1 2 T0.Next, let ~FFb=2�m be isomorphic to ~SS2

3. In this case we find j ¼ j0 ¼ 1 and i 6¼ i0.This implies xixi0 � x1x1 2 T0, and evidently, xjxj0 � xixi0 2 Tb=2.

Finally, assume ~FFb=2�m be isomorphic to ~SS13. Thus we have to consider the

subdigraph containing the arcs (2,1) and (1,3). Inspection gives usxjxj0 � xixi0 ¼ x1ðx3 � x2Þ. However both, x1 and x3 � x2 lie in T0. So we find againxjxj0 � xixi0 2 T0. This completes the discussion of possible cases, and concludesthe proof. h


Finally, we record a smiliar result for even prime powers. However, this timewe must not distinguish between several classes of even prime powers.

Lemma 13. Let ~HH be a simple digraph with non-negative cyclomatic number. Thereexists a constant q13 ¼ q13ð~HHÞ such that every even prime power q q13 with q � 1mod eð~HHÞ lies in I~HH .

Proof. The argument of Lemma 9 can easily be transferred to the case of evenprime powers. In particular, we have to use Lemma 5 instead of Lemma 4 toensure the existence of an appropriate k-tuple x. Furthermore, the essential ob-servation that �1 lies in the subgroup T is trivial for even q: in fields of even orderwe always have �1 ¼ 1. h

Summing up the results from this section, we obtain the following observation.

Corollary 14. For every simple digraph ~HH with non-negative cyclomatic number theset I~HH is non-empty and different from f1g.

Proof. Let q14 ¼ q14ð~HHÞ be the supremum of the constants q9; q12 and q13. Byvirtue of Dirichlet’s theorem there exists a prime p larger than q14 such that q � 1mod eð~HHÞ holds. However, exactly one of the existence lemmas stated in thissection applies to q. Thus I~HH contains q and satisfies the claim.

4. PBD–Closure

The above discussion ensures the existence of prime powers in the set I~HH ofadmissible values. In order to study the set I~HH in greater detail we require theconcept of pairwise balanced designs.

Consider a positive integer n and a subset K of the positive integers. A pairwisebalanced design PBD(n,K) is a pair ðV ;FÞ consisting of an n-set V and a col-lection F of subsets F � V (called blocks) such that every pair of distinct elementslies 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 nadmitting a pairwise balanced design PBDðn;KÞ. Consequently, the set K is said tobe PBD-closed iff K equals its PBD-closure. For a rigorous treatment of pairwisebalanced designs and related topics, the reader should consult e.g. [3].

Examining the set I~HH for a given digraph ~HH, we obtain the following result.

Lemma 15. For every simple digraph ~HH, the set I~HH is PBD-closed.

Proof. According to Lemma 1, we may assume that ~HH has non-negative cyclo-matic number. Let n be an integer from the PBD-closure of I~HH . Thus we find apairwise balanced design ðV ;FÞ of order n and block sizes from I~HH . For everyblock F ¼ fx1; . . . ; xjF jg 2 F, let GF ¼ f~GGF

x1; . . . ; ~GG

FxjF j

g be an idempotent

orthogonal decomposition of ~DDjF j by an ~HH-family ~GGF

of size ðjF j � 1Þ=eð~HHÞ.

298 S. Hartmann

For brevity, the set of all blocks F 2 F containing a fixed element x is said tobe the flower on x in F and denoted by FðxÞ. Let ~GGx be the union of all thedigraphs ~GG

Fx with F 2 FðxÞ. Obviously, ~GGx is again idempotent and an ~HH-family

of size ðn� 1Þ=eð~HHÞ. Thus, ~GGx has just eð~GGxÞ ¼ n� 1 arcs.Due to the definition of pairwise balanced designs, every pair x; y of distinct

elements from V belongs to exactly one block F 2 F. In GF there exists preciselyone digraph ~GG

Fz containing the arc ðx; yÞ. Hence, ~GGz is the only page in G con-

taining ðx; yÞ and, consequently, G is a decomposition of ~DDn.Furthermore, let ~GGx and ~GGy be two distinct digraphs in G. There is exactly one

block F in F containing both, x and y. Since GF is an orthogonal decomposition,

the union of ~GGFx and ~GG

Fy contains a digon. Due to our construction, this digon lies

in the union of ~GGx and ~GGy , too. This shows that the union of any two distinctpages in G contains at least (and thus exactly) one digon. h

A set K of positive integers is said to be eventually periodicwith period p iff, withsome integer r 2 K all sufficiently large integers n � r mod p lie inK, too. The notionof eventual periodic sets was introduced by Wilson [17] who proved the followingcriterion which is among the most interesting general theorems in design theory.

Theorem 16 (Wilson). Let K be a non-empty PBD-closed set different from f1g.Then K is eventually periodic with period bðKÞ ¼ g:c:dfnðn� 1Þ : n 2 Kg. In ad-dition, K has period p ¼ bðKÞ=2 iff p is odd and p � 0 mod aðKÞ holds, whereaðKÞ ¼ g:c:d:fn� 1 : n 2 Kg.

On applying Wilson’s theorem to the set I~HH we are able to verify the followingproperty.

Lemma 17. For every simple digraph ~HH with non-negative cyclomatic number, theset I~HH is eventually periodic with period eð~HHÞ.

Proof. The eventual periodicity of I~HH follows immediately from Lemma 15 andWilson’s theorem. It remains to determine a period of I~HH . By Lemma 1, we haven� 1 � 0 mod eð~HHÞ for every n 2 I~HH . Hence, bðI~HHÞ is divisible by eð~HHÞ. Suppose,eð~HHÞ is even. By Lemma 8 we find two primes p; q q14 with p � q � 1 modeð~HHÞ=2 and g:c:d:fpðp � 1Þ; qðq� 1Þg ¼ eð~HHÞ. Both, p and q are odd and relativelyprime to eð~HHÞ=2. Therefore, eð~HHÞ divides p � 1 as well as q� 1. It turns out thatp; q 2 I~HH and bðI~HHÞ ¼ eð~HHÞ holds.

We now come to the case in which eð~HHÞ is odd. This time, bðI~HHÞ is divisible by2eð~HHÞ. Again by Lemma 7, there are primes p; q 2 I~HH with g:c:d:fpðp � 1Þ;qðq� 1Þg ¼ 2eð~HHÞ such that bðI~HHÞ ¼ 2eð~HHÞ. Moreover, aðI~HHÞ is divisible by

eð~HHÞ, too. Hence aðI~HHÞ is either eð~HHÞ or 2eð~HHÞ depending on the existence of evenintegers in I~HH .

In order to find even integers in I~HH , consider the Euler totient function /ðnÞproviding the number of positive integers m n with g:c:d:fm; ng ¼ 1. Since eð~HHÞis odd, we obtain

2/ðeð~HHÞÞ � 1 mod eð~HHÞ


due to the Euler-Fermat Theorem (see e.g. [11]). By Lemma 13 there is a positiveinteger k with 2k/ðeð~HHÞÞ 2 I~HH . Thus we obtain aðI~HHÞ ¼ eð~HHÞ. Finally, Wilson’stheorem settles our claim. (

We are now in the position to establish our main result that the set I~HHcontains almost all integers n satisfying the necessary conditions given by Lemma1. We record this as follows:

Theorem 18. Let ~HH be a simple digraph with non-negative cyclomatic number.There exists a constant n18 ¼ n18ð~HHÞ such that for every n n18 with n � 1 modeð~HHÞ there is an idempotent orthogonal decomposition of ~DDn by an ~HH-family.

5. Conclusions

Let ~HH be a simple digraph with vð~HHÞ vertices and eð~HHÞ arcs. In the precedingsections we proved the existence of an orthogonal decomposition by an ~HH-familyfor all sufficiently large complete digraphs whenever eð~HHÞ vð~HHÞ holds. On theother hand, it is easy to check similar to Lemma 2 that there is no orthogonaldecomposition by ~HH-families if eð~HHÞ vð~HHÞ � 2 . The remaining case, namelyeð~HHÞ ¼ vð~HHÞ � 1 is of special interest. Here an orthogonal decomposition by an ~HH-family will only exist if this family is the digraph ~HH itself. As pointed out inSection 1, this problem is unsolved in general, and the methods used in this articledo not apply to this class of digraphs. Fortunately, the portion of simple digraphswith cyclomatic number -1 is considerably small.

Theorem 19. For almost every simple digraph ~HH there exists a constantm19 ¼ m19ð~HHÞ such that for every m m19 there is an orthogonal decomposition bym~HH.

Proof. Suppose ~HH has non-negative cyclomatic number. Choose m19 such thatm19eð~HHÞ þ 1 n18 holds. By Theorem 18 there exists an orthogonal decomposi-tion of ~DDn by m~HH if n ¼ með~HHÞ þ 1 n18, i.e. m m19 holds.

It remains to show that almost all simple digraphs have non-negative cyclo-matic number. Let uðn; bÞ and lðn; bÞ denote the numbers of unlabeled and labeledsimple digraphs with n vertices and b arcs, respectively. Moreover, let uðnÞ andlðnÞ be the total numbers of unlabeled and labeled simple digraphs with n vertices.It is easily verified, that

lðn; bÞ ¼n2

�b

0@

1A2b

and

lðnÞ ¼X1b¼0

lðn; bÞ ¼ 3

n2

� �

300 S. Hartmann

hold. In addition, it is well-known that almost every simple digraph has trivialautomorphism group, i.e. the proportion n!uðnÞ=lðnÞ tends to 1 for n ! 1 (cf. [4]).Finally, let u�ðnÞ and l�ðnÞ denote the numbers of unlabeled and labeled simpledigraphs with n vertices and negative cyclomatic number, respectively. We have

l�ðnÞ ¼Xn�1

b¼0

lðn; bÞ ¼Xn�1

b¼0

n2

�b

0@

1A2b

Xn�1

b¼0

n2

�b

2b n2n ¼ 32nlog3n:

This implies

0 u�ðnÞuðnÞ l�ðnÞ

uðnÞ ¼ n!l�ðnÞlðnÞ

lðnÞn!uðnÞ 33n log3n� n

2ð Þ lðnÞn!uðnÞ ¼ oð1Þ lðnÞ

n!uðnÞ

and consequently u�ðnÞ=uðnÞ tends to 0 for n ! 1. This concludes the proof. h

The statement of Theorem 19 is remarkable since it is widely believed thatthere exists an orthogonal decomposition of a complete digraph by almost allsimple digraphs. Hence, the constant m19ð~HHÞ is expected to be one for almost all~HH. However, a positive answer to this conjecture requires a proof which does notuse Wilson’s theorem on eventually periodic sets.

References

1. Alspach, B., Heinrich, K., Liu, G.: Orthogonal factorizations of graphs. In: J.H. Dinitz,D.R. Stinson: Contemporary Design Theory, chap. 2, pp. 13–40, New York: J. Wiley1992

2. Bennett, F.E., Wu, L.: On minimum matrix representation of closure operations.Discrete Appl. Math. 26, 25–40 (1990)

3. Beth, T., Jungnickel, D., Lenz, H.: Design Theory. Mannheim: BI 19854. Cameron, P.J.: Automorphism groups of graphs. In: L.W. Beineke and R.J. Wilson:

Selected topics in graph theory, vol. 2, pp. 89–127, London: Academic Press 19835. Demetrovics, J., Furedi, Z., Katona, G.O.H.: Minimum matrix representations of

closure operations. Discrete Appl. Math. 11, 115–128 (1985)6. Demetrovics, J., Katona, G.O.H.: Extremal combinatorial problems in relational

database. In: Fundamentals of computation theory, Lect. Notes. Comput. Sci. vol. 117,pp. 110–119 Berlin: Springer 1981

7. Ganter, B., Gronau, H.-D.O.F.: On two conjectures of Demetrovics, Furedi andKatona on partitions. Discrete Math. 88, 149–155 (1991)

8. Granville, A., Gronau, H.-D.O.F. Mullin, R.C.: On a problem of Hering concerningorthogonal double covers of Kn. J. Comb. Theory, Ser. A 72, 345–350 (1995)

9. Hering, F.: Balanced pairs. Ann. Discrete Math. 20, 177–182 (1984)10. Hering, F., Rosenfeld, M.: Problem number 38. In: K. Heinrich: Unsolved Problems:

Summer Research Workshop in Algebraic Combinatorics. Simon Fraser University1979

11. Parshin, A.N., Shafarevich, I.R.: Number Theory. Berlin: Springer 199012. Rausche, A.: On the existence of special block designs. Rostocker Math. Kolloq. 35,

13–20 (1988)13. Lidl, R. Niederreiter, H.: Finite fields. Cambridge: Cambridge University Press 1987


14. Schmidt, W.M.: Equations over finite fields. Berlin: Springer 197615. Weil, A.: On some exponential sums. Proc. Natl. Acad. Sci. USA, 34, 204–207 (1948)16. Wilson, R.M.: Cyclotomy and difference families in elementary abelian groups.

J. Number Theory, 4, 17–42 (1972)17. Wilson, R.M.: An existence theory for pairwise balanced designs II. J. Comb. Theory,

A 13, 246–270 (1972)18. Wilson, R.M.: Construction and uses of pairwise balanced designs. In: M. Hall Jr., J.H.

van Lint: Combinatorics, Math. Centre Tracts, pp. 18–41, Amsterdam: MathematischCentrum 1974

19. Wilson, R.M.: An existence theory for pairwise balanced designs III. J. Comb. Theory,Ser. A 18, 71–79 (1975)

Received: February 5, 1999Final version received: November 1, 1999

302 S. Hartmann

Documents

Orthogonal Decompositions of Complete Digraphs