THE SUM NUMBER OF d-PARTITE COMPLETE HYPERGRAPHS

Graph Theory 19 (1999 ) 79–91

THE SUM NUMBER OF d-PARTITE COMPLETE HYPERGRAPHS

Hanns-Martin Teichert

Institute of Mathematics, Medical University of L¨ ubeck Wallstraße 40, 23560 L¨ ubeck, Germany

Abstract

A d-uniform hypergraph H is a sum hypergraph iff there is a finite S ⊆ IN ⁺ such that H is isomorphic to the hypergraph H ⁺ _d (S) = (V, E), where V = S and E = {{v 1 , . . . , v d } : (i 6= j ⇒ v i 6= v j )∧ P d

i=1 v i ∈ S}.

For an arbitrary d-uniform hypergraph H the sum number σ = σ(H) is defined to be the minimum number of isolated vertices w 1 , . . . , w σ 6∈ V such that H ∪ {w 1 , . . . , w σ } is a sum hypergraph.

In this paper, we prove

σ(K ^d _n

_,...,n

) = 1 +

d

X

i=1

(n i − 1) + min (

0,

&

1 2

d−1

X

i=1

(n i − 1) − n d

!') ,

where K ^d n

,...,n

denotes the d-partite complete hypergraph; this gener- alizes the corresponding result of Hartsfield and Smyth [8] for complete bipartite graphs.

Keywords: sum number, sum hypergraphs, d-partite complete hypergraph.

1991 Mathematics Subject Classification: 05C65, 05C78.

1. Introduction and Definitions

The concept of sum graphs and integral sum graphs was introduced by Harary ([6], [7]). Many results for these kinds of graphs have been ob- tained in recent years, for a brief summary see for instance Sonntag and Teichert [12].

The graph theoretic concept mentioned above can be generalized to uni-

form hypergraphs as follows. All hypergraphs considered here are supposed

to be nonempty and finite without loops and multiple edges. In standard terminology we follow Berge [1].

A hypergraph H = (V, E) with vertex set V and edge set E ⊆ P(V )−{∅}

is d-uniform iff 2 ≤ d ∈ IN and |e| = d (∀e ∈ E). Let S ⊆ IN ⁺ be finite.

H ⁺ _d (S) = (V, E) is called the d-uniform sum hypergraph of S iff V = S and E =

(

{v 1 , v 2 , . . . , v _d } : (i 6= j ⇒ v i 6= v j ) ∧

d

X

i=1

v i ∈ S )

.

The d-uniform hypergraph H is a sum hypergraph iff there exists a set S ⊆ IN ⁺ such that H ∼ = H ⁺ _d (S). For d = 2 we obtain the known concept of sum graphs. For an arbitrary d-uniform hypergraph H the sum number σ = σ(H) is defined to be the minimum number of isolated vertices w 1 , . . . , w σ 6∈ V such that H ∪{w 1 , . . . , w _σ } is a sum hypergraph. If also nonpositive integers are allowed as elements of S, i.e. S ⊆ ZZ, we obtain the definitions of integral sum hypergraphs and the integral sum number ζ = ζ(H) in the same manner.

As for graphs, the determination of the sum number (integral sum num- ber) for certain classes of hypergraphs is an interesting question. The fol- lowing results are known:

• If T d denotes a d-uniform hypertree, then d ≥ 3 implies σ(T _d ) = 1 and ζ(T _d ) = 0 (Sonntag and Teichert ([12], [13])). Ellingham [5] proved for nontrivial trees T 2 = T that σ(T ) = 1. Sharary [11] showed that all cater- pillars are integral sum graphs and Chen [4] proved that the generalized stars and the trees in which any two distinct forks have distance at least four are integral sum graphs; both authors conjecture ζ(T ) = 0 for all trees but this problem remains still open.

• For the d-uniform complete hypergraph on n vertices K ^d _n we obtain the sum number σ(K ^d _n ) = d(n − d) + 1 for n − 2 ≥ d ≥ 2; this was shown by Sonntag and Teichert [13] for d ≥ 3 and by Bergstrand et al. [2]

for graphs K ² _n = K n . Chen [3] as well as Sharary [10] showed that for complete graphs ζ(K n ) = σ(K n ) if n ≥ 4. For n − 2 ≥ d ≥ 3 Sonntag and Teichert [13] found bounds for ζ(K ^d _n ) and conjectured that ζ(K ^d _n ) = σ(K ^d _n ) is true for hypergraphs too.

In this paper, we determine the sum number for a third class of uniform hy- pergraphs. As a generalization of complete bipartite graphs Berge [1] defined the d-partite complete hypergraph K ^d _n

_,...,n

d

as follows: Let X 1 , X 2 , . . . , X _d be

pairwise disjoint sets of cardinalities n 1 ≤ n 2 ≤ . . . ≤ n d . The vertices of

K ^d _n

_,...,n

are the elements of ^{S d} _i=1 X i and the edges are all {v 1 , v 2 , . . . , v _d }

with v _i ∈ X i for i = 1, . . . , d.

For x ∈ IR let dxe denote the smallest integer ≥ x. Hartsfield and Smyth [8] proved for complete bipartite graphs K _n

_,n

Theorem 1. For given integers n 1 ≥ 2 and n 2 ≥ n 1 holds σ(K n

,n

) =

 1

2 (3n 1 + n 2 − 3)

 .

For the symmetric bipartite graph K _n,n Miller et al. [9] showed Theorem 2. ζ(K _n,n ) = σ(K _n,n ) f or n ≥ 2.

The problem to determine ζ(K _n

_,n

) for n 1 6= n 2 remains still open.

In the following, we generalize Theorem 1. In Section 2, we prove several lemmata; for the determination of the sum number we distinguish two cases concerning the cardinality of the maximum vertex subset X _d of K ^d _n

_,...,n

d

. Summarizing these results, we give in Section 3 a general formula for the sum number of d-partite complete hypergraphs.

2. Two Cases for the Determination of the Sum Number We use the following notations:

X 1 , . . . , X d is the vertex partition of the complete d-partite hypergraph K ^d _n

_,...,n

, where n i denotes the cardinality of X i and n 1 ≤ n 2 ≤ . . . ≤ n d

is fulfilled. E is the set of edges of K ^d _n

_,...,n

and Y is a set of isolated ver- tices such that for some labelling K _n ^d

_,...,n

d

∪ Y can be recognized as a sum hypergraph. All vertices of ^{S d} _i=1 X _i ∪ Y are referenced by their labels.

Lemma 3. There are 1+ ^{P d} _i=1 (n _i −1) pairwise different sums v 1 +v 2 +. . .+v _d of vertices v i ∈ X i , i = 1, . . . , d.

P roof. Using the notation X i = {v ¹ _i , . . . , v ⁿ _i

} with v _i ^k < v _i ^l if k < l for i = 1, . . . , d we consider the following sets of sums:

S 1 = {v 1 ^j + v 2 ¹ + . . . + v ¹ _d : j ∈ {1, . . . , n 1 }},

S i = {v 1 ⁿ

+ . . . + v ⁿ _i−1

+ v ^j _i + v ¹ _i+1 + . . . + v ¹ _d : j ∈ {2, . . . , n i }}, i = 2, . . . , d.

Clearly, S i ∩ S j = ∅ for i 6= j and |S 1 | = n 1 , |S i | = n i − 1 for i = 2, . . . , d.

Hence there are 1 + ^{P d} _i=1 (n _i − 1) pairwise different sums.

Lemma 4. Consider a labelling of V ⁰ = ^{S d} _i=1 X i ∪ Y such that K ^d _n

_,...,n

∪ Y is a sum hypergraph and let v ^∗ _d ∈ X d be arbitrarily chosen. Then

( _d−1 X

i=1

v i + v _d ^∗ : v i ∈ X i

)

⊆ X d

!

∨

( _d−1 X

i=1

v i + v _d ^∗ : v i ∈ X i

)

⊆ Y

! (1)

P roof. 1. Suppose there are v _i ^∗ ∈ X i , i = 1, . . . , d − 1, such that ^{P d} _i=1 v ^∗ _i =

˜

v _d ∈ X d and let v _i ∈ X i i = 1, . . . , d − 1, be arbitrarily chosen. In part 1, we show that this implies v ⁰ _d := ^{P d−1} _i=1 v _i + v ^∗ _d ∈ X d .

Because of {v 1 , . . . , v _d−1 , ˜ v d } ∈ E there is a vertex

v ⁰⁰ =

d−1

X

i=1

v i + ˜ v _d =

d−1

X

i=1

v i +

d

X

i=1

v _i ^∗ =

d−1

X

i=1

v _i ^∗ + v ⁰ _d .

Hence {v ^∗ ₁ , . . . , v _d−1 ^∗ , v ⁰ _d } ∈ E and by the definition of the d-partite complete hypergraph it follows v ⁰ _d ∈ X d .

2. Suppose there are v _i ^∗ ∈ X i , i = 1, . . . , d − 1, such that ^{P d}

i=1

v ^∗ _i = ˜ v _j ∈ X j

for j ∈ {1, . . . , d − 1}. It follows analogously to part 1 that

∀ i ∈ {1, . . . , j − 1, j + 1, . . . , d} ∀ v i ∈ X i :

d

X

i=1 i6=j

v i + v _j ^∗ ∈ X j .

By Lemma 3 there are at least 1 + ^{P d}

(n i − 1) pairwise different sums each containing v _j ^∗ . With v ^∗ _j ∈ X j we obtain

n _j = |X _j | ≥ 1 +





 1 +

d

X

i=1 i6=j

(n _i − 1)





 ≥ 1 + n d ,

which contradicts the supposition n _j ≤ n d made in the beginning of this chapter. Hence this case is impossible.

3. Suppose there are v ^∗ _i ∈ X i , i = 1, . . . , d − 1, such that ^{P d} _i=1 v ^∗ _i ∈ Y. With

parts 1, 2 we obtain ^{P d−1} _i=1 v i + v _d ^∗ ∈ / ^{S d} _i=1 X i for arbitrary vertices v i ∈ X i ,

i = 1, . . . , d−1. By the sum hypergraph property we obtain ^{P d−1} _i=1 v i +v ^∗ _d ∈ Y

and this proves (1).

If ˆ V ⊆ ^{S d} _i=1 X i ∪ Y denotes the subset of vertices representing sums ^{P d} _i=1 v i

where {v 1 , . . . , v _d } ∈ E it follows from Lemma 4 that only the cases ˆ V = Y (case 1) or ˆ V ∩ X _d 6= ∅ (case 2) are possible. Clearly, it depends on the cardinality n _d of X _d whether the first case appears or the second one is possible.

Case 1. n _d ≤ 1 + ^{P d−1} _i=1 (n i − 1) (i.e., ˆ V = Y ).

Lemma 5. If n _d ≤ 1 + ^{P d−1} _i=1 (n _i − 1) then

σ(K ^d _n

_,...,n

) ≥ 1 +

d

X

i=1

(n i − 1).

(2)

P roof. Let v _d ^∗ ∈ X d be arbitrarily chosen and assume ^{P d−1} _i=1 v _i + v _i ^∗ ∈ X _d for some v _i ∈ X i , i = 1, . . . , d − 1. Then Lemma 4 implies that each sum containing v ^∗ _d belongs to X d and from Lemma 3 follows n d ≥ 1 +

 1 + ^{P d−1} _i=1 (n _i − 1) ^ , a contradiction. Thus ˆ V ⊆ Y and with Lemma 3 we obtain (2).

Our aim is to show that equality is fulfilled in (2). For this purpose we describe an appropriate labelling:

Lemma 6. Let ^{S d} _i=1 X i ∪ Y be labelled as follows:

X i = {x i + 1, . . . , x i + n i }, i = 1, . . . , d;

Y =

( _d X

i=1

(x i + 1), . . . ,

d

X

i=1

(x i + n i ) ) (3)

with x _i = 10 ^t+i ; i = 1, . . . , d, where t ≥ lg (dn _d ).

Then the resulting sum hypergraph consists of K _n ^d

_,...,n

with vertex set S _d

i=1 X _i and |Y | = 1 + ^{P d} _i=1 (n _i − 1) isolated vertices.

P roof. Clearly, (3) implies Y = ˆ V = ^{n P d} _i=1 v _i : v _i ∈ X i , i = 1, . . . , d ^o and

|Y | = 1 + ^{P d} _i=1 (n _i − 1).

It remains to show that the labelling does not induce too many edges by the sum hypergraph property, i.e., if ^{P d} _i=1 v ⁱ ∈ Y then {v ¹ , . . . , v ^d } ∈ E follows.

Let α 1 , . . . , α _d+t+1 be the digits of the label α 1 α 2 . . . α _d+t+1 of a vertex v.

The condition t ≥ lg (dn _d ) implies

• if v ∈ X i (i ∈ {1, . . . , d}) , then

α _d−i+1 = 1; α _j = 0 for j ∈ {1, . . . , d} − {d − i + 1},

• if v ∈ Y, then

α _j = 1 for j ∈ {1, . . . , d}.

(4)

Thus we have for arbitrary pairwise different vertices v ¹ , . . . , v ^d ∈ ^{S d} _i=1 X i ∪ Y that ^{P d} _i=1 v ⁱ ∈ X / i , i = 1, . . . , d. Further in case of ^{P d} _i=1 v ⁱ ∈ Y, neither v ^k , v ^l ∈ X i nor v ^k ∈ Y for some i, k, l ∈ {1, . . . , d}, k 6= l is possible, because this would imply α j 6= 1 for at least one j ∈ {1, . . . , d} in both of the cases, a contradiction to (4).

Summarizing the results of Lemmata 5, 6, we have shown Theorem 7. If n d ≤ 1 + ^{P d−1} _i=1 (n i − 1), then

σ(K _n ^d

_,...,n

) = 1 +

d

X

i=1

(n i − 1);

(5)

especially for n 1 = . . . = n _d = n we have

σ(K ^d _n,...,n ) = 1 + d(n − 1).

(6)

Notice that for complete bipartite graphs K _n

_,n

because of n 1 ≤ n 2 the supposition of Theorem 7 can only be true for n 1 = n 2 = n and this leads to σ(K _n,n ) = 2n − 1 as a special case of (6) which corresponds to the value given in Theorem 1.

Case 2. n d ≥ 2 + ^{P d−1} _i=1 (n i − 1) (i.e. ˆ V ∩ X d 6= ∅).

We introduce the notations

X _d ⁰ = {v _d ⁰ ∈ X d ∀ i ∈ {1, . . . , d − 1} ∃ v _i ⁰ ∈ X i :

d

X

i=1

v ⁰ _i ∈ X d }, (7)

X _d ⁰⁰ = {v ⁰⁰ _d ∈ X d ∀ i ∈ {1, . . . , d} ∃ v _i ⁰ ∈ X i : v _d ⁰⁰ =

d

X

i=1

v _i ⁰ }.

(8)

The aim of this section is the construction of a labelling that reduces the cardinality of Y (below the value given in (5)) to a minimum. Due to Lemma 4 this is only possible by the maximization of |X _d ⁰ |.

In the following, we suppose that at least two vertex subsets X _i contain two or more vertices, i.e.,

2 ≤ n _d−1 ≤ n d

(9)

(otherwise we obtain σ(K ^d _n

_,...,n

) = 1 immediately).

Lemma 8. If a labelling of K ^d _n

_,...,n

generates a maximum cardinality |X _d ⁰ |, then

X _d ⁰ ∩ X _d ⁰⁰ = ∅.

(10)

P roof. We prove (10) by induction on n d = |X d |. For n d ≤ 1+ ^{P d−1} _i=1 (n i −1) we have shown X _d ⁰ = ∅ in Case 1, hence (10) is true. Now suppose that (10) is valid for some n _d ≥ 2 + ^{P d−1} _i=1 (n _i − 1) and form ˜ X _d by adding one new vertex to X d . Then ˜ n d = | ˜ X d | = n d + 1, further we denote by ˜ X _d ⁰ ⊆ ˜ X d

and ˜ X _d ⁰⁰ ⊆ ˜ X _d the sets corresponding to (7) and (8), respectively. In the following, we have to show that ˜ X _d ⁰ ∩ ˜ X _d ⁰⁰ = ∅.

1. We show

|X _d ⁰ | ≤ | ˜ X _d ⁰ | ≤ |X _d ⁰ | + 1.

(11)

The first inequality is obvious. Let v _d ^min = min{v ⁰ _d : v ⁰ _d ∈ ˜ X _d ⁰ }, then v _d ^min ∈ ˜ / X _d ⁰⁰ .

(12)

Now assume that | ˜ X _d ⁰ | ≥ |X _d ⁰ | + 2 is fulfilled and consider ˜ X _d ⁰ − {v _d ^min }. Then

| ˜ X _d − {v _d ^min }| = n d and with (12) we obtain | ˜ X _d ⁰ − {v ^min _d }| ≥ |X _d ⁰ | + 1, a contradiction to the maximality of |X _d ⁰ |, hence (11) is true.

2. Now we prove

| ˜ X _d ⁰ | = |X _d ⁰ | ⇒ ˜ X _d ⁰ ∩ ˜ X _d ⁰⁰ = ∅.

(13)

With v _d ^max = max{v _d ⁰⁰ : v _d ⁰⁰ ∈ ˜ X _d ⁰⁰ } we obtain

v ^max _d ∈ ˜ / X _d ⁰ .

(14)

Consider ˆ X _d := ˜ X _d − {v _d ^max } and form the sets ˆ X _d ⁰ and ˆ X _d ⁰⁰ corresponding to (7) and (8) respectively. Then ˆ X _d ⁰ ⊆ ˜ X _d ⁰ and next we show that even equality is fulfilled:

Consider an arbitrary ˆ v d ∈ ˜ X _d ⁰ . Then there are ˆ v i ∈ X i , i = 1, . . . , d − 1, such that ^{P d} _i=1 v ˆ i = ¯ v _d ∈ ˜ X _d ⁰⁰ . If ¯ v _d 6= v ^max _d then ¯ v _d ∈ ˆ X _d ⁰⁰ and hence ˆ v _d ∈ ˆ X _d ⁰ . If ¯ v d = v ^max _d , we choose ˆˆ v i ∈ X i , i = 1, . . . , d − 1, and because of (9) we can suppose that ˆ v j 6= ˆˆ v j for exactly one j ∈ {1, . . . , d − 1}. Lemma 4 implies P _d−1

i=1 ˆˆv i + ˆ v d = ¯¯ v d ∈ ˜ X _d ⁰⁰ and because of ¯¯ v d 6= ¯ v d = v _d ^max we have ¯¯ v d ∈ ˆ X _d ⁰⁰ , hence ˆ v _d ∈ ˆ X _d ⁰ again.

Together this yields

X ˆ _d ⁰ = ˜ X _d ⁰ . (15)

From the left side of (13) follows | ˆ X _d ⁰ | = |X _d ⁰ |. Because of | ˜ X _d − {v ^max _d }| = n d

we obtain by the induction hypothesis ˆ X _d ⁰ ∩ ˆ X _d ⁰⁰ = ∅. Using (15) and (14) this leads to

∅ = ˆ X _d ⁰ ∩ ˆ X _d ⁰⁰ = ˜ X _d ⁰ ∩ ˆ X _d ⁰⁰ = ˜ X _d ⁰ ∩ ( ˆ X _d ⁰⁰ ∪ v _d ^max ) = ˜ X _d ⁰ ∩ ˜ X _d ⁰⁰ . 3. Because of (11) and (13) we can suppose

| ˜ X _d ⁰ | = |X _d ⁰ | + 1 (16)

in the following. Next we prove that

| ˜ X _d ⁰ ∩ ˜ X _d ⁰⁰ | ≤ 1 (17)

must be true.

Assume | ˜ X _d ⁰ ∩ ˜ X _d ⁰⁰ | ≥ 2 and let v _d ¹ , v _d ² ∈ ˜ X _d ⁰ ∩ ˜ X _d ⁰⁰ be two distinct vertices.

Further let v _d ^min ∈ ˜ X _d ⁰ − ˜ X _d ⁰⁰ be defined as in part 1 of the proof and consider X ˇ _d = ˜ X _d − {v _d ^min } with the subsets ˇ X _d ⁰ and ˇ X _d ⁰⁰ formed corresponding to (7) and (8), respectively. Then | ˇ X d | = n d and (16) implies | ˇ X _d ⁰ | = |X _d ⁰ |. Using the induction hypothesis we obtain ˇ X _d ⁰ ∩ ˇ X _d ⁰⁰ = ∅, i.e., the deletion of v _d ^min in ˜ X _d ⁰ causes that v _d ¹ , v _d ² ∈ ˇ / X _d ⁰⁰ . Hence

∀ i ∈ {1, . . . , d − 1} ∃ v _i ¹ , v _i ² ∈ X i :

d−1

X

i=1

v ¹ _i + v _d ^min = v _d ¹ ∧

d−1

X

i=1

v ² _i + v _d ^min = v _d ²

(18)

and

∀ i ∈ {1, . . . , d − 1} ∀ v _i ⁰ ∈ X i ∀v _d ⁰ ∈ ˜ X _d ⁰ : v ⁰ _d 6= v ^min _d ⇒

d

X

i=1

v ⁰ _i 6= v ¹ _d ∧

d

X

i=1

v _i ⁰ 6= v _d ²

! . (19)

These facts are useful for the consideration of X _d = ˜ X _d − {v _d ² } with the subsets X ⁰ _d and X ⁰⁰ _d formed corresponding to (7) and (8), respectively: Using (18) and (19) we obtain v ¹ _d ∈ X ⁰⁰ _d and v _d ¹ ∈ X ⁰ _d , respectively. Hence

v _d ¹ ∈ X ⁰ _d ∩ X ⁰⁰ _d . (20)

On the other hand, we obtain from (18) and (19) that v _d ^min ∈ X ⁰ _d and X ˜ _d ⁰ − {v ² _d } ⊆ X ⁰ _d , respectively, i.e., X ⁰ _d = ˜ X _d ⁰ − {v ² _d } and with (16) we obtain

|X ⁰ _d | = |X _d ⁰ |. Because of |X _d | = |X d | = n d the induction hypothesis yields X ⁰ _d ∩ X ⁰⁰ _d = ∅; a contradiction to (20) which implies the validity of (17).

4. We conclude the proof by constructing a contradiction to (17) in case of X ˜ _d ⁰ ∩ ˜ X _d ⁰⁰ 6= ∅. Suppose ˆ v _d ∈ ˜ X _d ⁰ ∩ ˜ X _d ⁰⁰ , then

∀ i ∈ {1, . . . , d − 1} ∃ v _i ⁰ ∈ X i ∃ v ⁰ _d ∈ ˜ X _d ⁰ : ˆ v d =

d

X

i=1

v ⁰ _i . (21)

Consider vertices v _i ¹ , v ² _i ∈ X i for i = 1, . . . , d. By (9) we can suppose v ¹ _k 6= v _k ² for at least one k ∈ {1, . . . , d − 1}. Using Lemma 4 and ˆ v _d ∈ ˜ X _d ⁰ we obtain P _d−1

i=1 v ^j _i + ˆ v _d ∈ ˜ X _d ⁰⁰ , j = 1, 2 and with (21) this yields

d−1

X

i=1

v _i ^j +

d

X

i=1

v ⁰ _i =

d−1

X

i=1

v ^j _i + v ⁰ _d

! +

d−1

X

i=1

v _i ⁰ ∈ ˜ X _d ⁰⁰ for j = 1, 2.

Hence

d−1

X

i=1

v ^j _i + v ⁰ _d ∈ ˜ X _d ⁰ , j = 1, 2.

(22)

On the other hand, v _d ⁰ ∈ ˜ X _d ⁰ implies ^{P d−1} _i=1 v _i ^j + v ⁰ _d ∈ ˜ X _d ⁰⁰ for j = 1, 2 and

together with (22) it follows | ˜ X _d ⁰ ∩ ˜ X _d ⁰⁰ | ≥ 2; a contradiction to (17). Hence

X ˜ _d ⁰ ∩ ˜ X _d ⁰⁰ = ∅ and the proof of Lemma 8 is completed.

Lemma 9. If n _d ≥ 2 + ^{P d−1} _i=1 (n i − 1), then

σ(K ^d _n

_,...,n

) ≥

&

1 2 3

d−1

X

i=1

(n i − 1) + n d

!' . (23)

P roof. As mentioned before, we have to maximize n ⁰ _d = |X _d ⁰ | to obtain a minimum number |Y | of isolated vertices. Lemma 8 yields X _d ⁰⁰ ⊆ X d − X _d ⁰ and with Lemmata 3, 4 we obtain n _d − n ⁰ _d ≥ n ⁰ _d + ^{P d−1} _i=1 (n _i − 1), i.e.,

n ⁰ _d ≤ 1

2 n _d −

d−1

X

i=1

(n _i − 1)

! . (24)

Again using Lemma 3 we obtain with (24) a bound for the number of dif- ferent sums containing elements of X 1 , . . . , X _d−1 , X _d − X _d ⁰ :

|Y | ≥ 1 +

d−1

X

i=1

(n i − 1) + (n d − n ⁰ _d − 1) ≥ 1 2 3

d−1

X

i=1

(n i − 1) + n d

! ,

hence (23) is fulfilled.

As in case 1 we next describe a labelling that provides equality in (23).

First of all suppose that n _d − ^{P d−1} _i=1 (n _i − 1) is even and choose n ⁰ _d =

1

2 (n d − ^{P d−1} _i=1 (n i − 1)) which is by (24) the maximum possible value. Now let ^{S d} _i=1 X i ∪ Y be labelled as follows:

X _i = {x _i + 1, . . . , x _i + n _i } , i = 1, . . . , d − 1, X _d ⁰ = {x d + 1, . . . , x d + n ⁰ _d },

X _d ⁰⁰ = ( _d

X

i=1

(x _i + 1), . . . ,

d−1

X

i=1

(x _i + n _i ) + x _d + n ⁰ _d )

, (25)

Y = ( _d−1

X

i=1

(x _i + 1) +

d

X

i=1

(x _i + 1), . . . ,

d−1

X

i=1

(x _i + n _i ) +

d−1

X

i=1

(x _i + n _i ) + x _d + n ⁰ _d )

with x _i = 10 ^t+i , i = 1, . . . , d, where t ≥ lg ((2d − 1)n _d ).

A simple calculation yields |X _d ⁰ | + |X _d ⁰⁰ | = |X d | and |Y | = ¹ ₂ (3 ^{P d−1} _i=1 (n i − 1)

+n _d ). The labelling for odd values of n _d − ^{P d−1} _i=1 (n _i − 1) can be obtained

from (25) (constructed for n _d + 1 vertices) by deleting any vertex of X _d ⁰ . For this case follows |Y | = ¹ ₂ (3 ^{P d−1} _i=1 (n _i − 1) + n d + 1).

Summarizing the results we obtain:

|Y | =

&

1 2 3

d−1

X

i=1

(n _i − 1) + n d

!' . (26)

Similarly to case 1, Lemma 6, we will show

Lemma 10. If ^{S d} _i=1 X _i ∪ Y is labelled according to (25) the resulting sum hypergraph consists of K ^d _n

_,...,n

with vertex set ^{S d} _i=1 X i and |Y | isolated vertices.

P roof. Clearly the labelling generates the edge set E of K ^d _n

_,...,n

d

and we have to show that (25) does not induce too many edges by the sum hypergraph property. Again let α 1 α 2 . . . α _d+t+1 be a vertex with digits α 1 , . . . , α _d+t+1 . Because of t ≥ lg((2d − 1)n _d ) we obtain three types of ver- tices:

• for any vertex of X ˜ i =

( X _i , for i ∈ {1, . . . , d − 1}, X _d ⁰ , for i = d :

α _d−i+1 = 1 ; α j = 0 for j ∈ {1, . . . , d} − {d − i + 1}, (27)

• for any vertex of X _d ⁰⁰ : α j = 1 for j ∈ {1, . . . , d},

• for any vertex of Y : α 1 = 1 ; α _j = 2 for j ∈ {2, . . . , d}.

Let s = ^{P d} _i=1 v ⁱ be a sum of pairwise disjoint vertices of ^{S d} _i=1 X i ∪ Y. We use the notation S = {v ¹ , . . . , v ^d } and distinguish the following cases:

(A) |S ∩ Y | = 1, (B) |S ∩ Y | ≥ 2, (C) S ∩ Y = ∅,

(C1) |S ∩ X _d | ≥ 2,

(C2) |S ∩ X _d | ≤ 1 ∧ ∃ m ∈ {1, . . . , d − 1} : |S ∩ X m | ≥ 2, (C21) S ∩ X _d ⁰⁰ 6= ∅,

(C22) S ∩ X _d ⁰⁰ = ∅.

Using (27) it can easily be seen that for each of these cases the following holds:

α 1 ≥ 2

or α i ≥ 3 for at least one i ∈ {2, . . . , d}

or α _i = 0, α _j ≥ 2 for some i, j ∈ {2, . . . , d},

i.e., s / ∈ ^{S d} _i=1 X i ∪ Y. Hence s ∈ ^{S d} _i=1 X i ∪ Y iff {v ¹ , . . . , v ^d } ∈ E.

Summarizing the results of Lemmata 8, 9 and (26), we have shown Theorem 11. If n d ≥ 2 + ^{P d−1} _i=1 (n i − 1), then

σ(K ^d _n

_,...,n

) =

&

1 2 3

d−1

X

i=1

(n i − 1) + n d

!' . (28)

3. Main Result In case of n _d ≥ 2 + ^{P d−1} _i=1 (n _i − 1) we obtain with (28)

σ(K ^d _n

_,...,n

) = 1 +

d

X

i=1

(n i − 1)

! +

&

1 2

d−1

X

i=1

(n i − 1) − n _d

!' .

Here the first summand is the sum number (5) for the case n _d ≤ 1 + P _d−1

i=1 (n _i − 1) given in Theorem 7 and the second one is negative for n _d ≥ 2 + ^{P d−1} _i=1 (n i − 1) but nonnegative for n _d ≤ 1 + ^{P d−1} _i=1 (n i − 1). This yields the main result:

Theorem 12. For d ≥ 2 and 2 ≤ n _d−1 ≤ n d the sum number of the d-partite complete hypergraph is given by

σ(K ^d _n

1

,...,n

) = 1 +

d

X

i=1

(n _i − 1) + min (

0,

&

1 2

d−1

X

i=1

(n _i − 1) − n d

!') .

Obviously, for d = 2, because of (n 1 − 1) − n 2 ≤ −1 we obtain the value given in Theorem 1.

Acknowledgement

The author wishes to thank the referee for his valuable suggestions.

References

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Received 24 August 1998

Revised 4 January 1999