CLASSES OF HYPERGRAPHS WITH SUM NUMBER ONE

Graph Theory 20 (2000 ) 93–103

CLASSES OF HYPERGRAPHS WITH SUM NUMBER ONE

Hanns-Martin Teichert Institute of Mathematics Medical University of L¨ubeck Wallstraße 40, 23560 L¨ubeck, Germany

Abstract

A hypergraph H is a sum hypergraph iff there are a finite S ⊆ IN

and d, d ∈ IN

with 1 < d ≤ d such that H is isomorphic to the hypergraph H

(S) = (V, E) where V = S and E = {e ⊆ S : d ≤

|e| ≤ d ∧ P

v∈e

v ∈ S}. For an arbitrary hypergraph H the sum number σ = σ(H) is defined to be the minimum number of isolated vertices w

, . . . , w

6∈ V such that H ∪ {w

, . . . , w

} is a sum hypergraph.

For graphs it is known that cycles C

and wheels W

have sum numbers greater than one. Generalizing these graphs we prove for the hypergraphs C

and W

that under a certain condition for the edge cardinalities σ(C

) = σ(W

) = 1 is fulfilled.

Keywords: hypergraphs, sum number, vertex labelling.

1991 Mathematics Subject Classification: 05C65, 05C78.

1 Introduction and Definitions

The notion of sum graphs was introduced by Harary [4]. This graph theoretic concept can be generalized to hypergraphs as follows.

All hypergraphs considered here are supposed to be nonempty and fi-

nite, without loops and multiple edges. In standard terminology we follow

Berge [1]. By H = (V, E) we denote a hypergraph with vertex set V and

edge set E ⊆ P(V )\{∅}. Further we use the notations d = d(H) = min

{|e| : e ∈ E} and d = d(H) = max {|e| : e ∈ E}; if d = d = d, we say H is

a d-uniform hypergraph. A hypergraph is linear if no two edges intersect in

more than one vertex.

Let S ⊆ IN

be finite and d, d ∈ IN

such that 1 < d ≤ d. Then H

(S) = (V, E) is called a (d, d)-sum hypergraph of S iff V = S and E = {e ⊆ S : d ≤ |e| ≤ d ∧ ^P

v∈e

v ∈ S}. Furthermore, a hypergraph H is a sum hypergraph iff there exist S ⊆ IN

and d, d ∈ IN

such that H is isomorphic to H

(S).

For d = d = 2 we obtain the known concept for graphs. For an arbitrary hypergraph H the sum number σ = σ(H) is defined to be the minimum number of isolated vertices w

, . . . , w

6∈ V such that H ∪ {w

, . . . , w

} is a sum hypergraph.

Generalizing known results for classes of graphs, the sum number for the corresponding classes of hypergraphs was determined in the following three cases:

• Trees T : σ(T ) = 1 (Ellingham [3]);

hypertrees T with d < 2d − 1 : σ(T ) = 1 (Sonntag and Teichert [9]).

• Complete graphs K

with n ≥ 4 : σ(K

) = 2n − 3 (Bergstrand et al. [2]);

d-uniform complete hypergraphs K

with n ≥ d + 2 : σ(K

) = d(n − d) + 1 (Sonntag and Teichert [10]).

• Complete bipartite graphs K _l

with 2 ≤ n

≤ n

: σ(K

) =

12

(3n

+ n

− 3) ^m (Hartsfield and Smyth [6]);

d-partite complete hypergraphs K

with 2 ≤ n

≤ n

: σ(K

) = 1 + ^P

i=1

(n

− 1)+ min (

0,

&

12

Ã

P

i=1

(n

− 1) − n

!')

(Teichert [11]).

In this paper, we determine the sum number for two other classes of hyper- graphs. As a generalization of cycles C

and wheels W

, n ≥ 3, we obtain the linear hypergraphs C

and W

, respectively. These hypergraphs have the same number of edges as the corresponding graphs, but now each edge e

consists of an arbitrary number d

≥ 2 of vertices. Note that any two edges of C

and W

, respectively, have only a vertex in common if this is explicitly claimed in the following definitions. In detail, a hypercycle C

= (V

, E

) is defined by

V

= ^S

i=1

{v

, . . . , v

},

E

= {e

, . . . , e

} with e

= {v

, . . . , v

= v

} (1)

where i + 1 is taken mod n.

A hyperwheel W

= (V

, E

) is defined by V

= V

∪ {c} ∪ ^S

i=1

{v

, . . . , v

}, E

= E

∪ {e

, . . . , e

} (2)

with e

= {v

= c, v

, . . . , v

n+i−1

, v

n+i

= v

}.

The edges of E

are called the rim of the hyperwheel, the vertex c is the centre and the edges e

, . . . , e

are the spokes of W

. Obviously, for d

= 2, i = 1, . . . , n and i = 1, . . . , 2n it follows C

= C

and W

= W

, respectively.

The sum numbers of cycles (Harary [5]) and wheels (Hartsfield and Smyth [7], Miller et al. [8]) are known:

σ(C

) =

( 2, if n 6= 4,

3, if n = 4. σ(W

) = (

2

+ 2, if n even and n ≥ 4, n, if n odd and n ≥ 5.

Now observe that both, trees and hypertrees, have sum number one and that the sum number for complete graphs and complete bipartite graphs can be obtained from the more general formula for the corresponding hypergraphs by setting d = 2. In contrast to these observations, we show in the following that under a certain condition for the edge cardinalities σ(C

) = σ(W

) = 1 is fulfilled.

2 An Algorithm for Labelling Hypercycles and Hyperwheels

The algorithm given below is a modification of an algorithm for labelling the vertices of hypertrees (Sonntag, Teichert [9]). It starts with labelling the vertices of e

, . . . , e

, therefore it can be used for labelling both, C

and W

.

Let ˜ v / ∈ V

be an isolated vertex, ˜ C

= C

∪ {˜ v} and ˜ W

= W

∪ {˜ v}.

We consider the vertex set V ( ˜ H

) of the hypergraph ˜ H

∈ { ˜ C

, ˜ W

} and construct a labelling r : V ( ˜ H

) → IN

. This vertex labelling r induces the mapping r

:

P(V ( ˜ H

)) 3 M 7→ r

(M ) = ^X

v∈M

r(v) ∈ IN

.

In the following assume

d( ˜ H

) ≥ 3.

(3)

Before formulating the algorithm in detail we give a summary of the main steps: In the algorithm the vertices are labelled edge by edge. Whenever a vertex gets its label we add it to the set L of labelled vertices.

– We start with initialization (i.e., labelling the vertices of e

similarly as described in the next step).

– If e

, 1 ≤ i ≤ n−1, is completly labelled but e

not then r(v

) = r(v

) is given, v

is labelled by the sum of the labels of all vertices of e

. The other nonlabelled vertices of e

get labels greater than the sum of all vertices of L.

– The value r

(e

) is assigned to ˜ v (if ˜ H

= ˜ C

) or c (if ˜ H

= ˜ W

). In the first case we are done; in the other case we label the vertices v

, . . . , v

of the spokes e

, . . . , e

similarly as described above and assign r

(e

) to ˜ v.

We use the notations introduced in (1) and (2) for C

and W

, respectively.

ALGORITHM: Let ˜ H

∈ { ˜ C

, ˜ W

} be a hypergraph containing the iso- lated vertex ˜ v.

for i = 1 to n do

if i = 1 then [r(v

) := 1; L := {v

}; α := 10];

r(v

) := α; L := L ∪ {v

};

if i < n then k := d

else k := d

− 1;

for j = 3 to k do

r(v

) := 10 · r

(L); L := L ∪ {v

};

end for;

α := r

(e

);

end for;

if ˜ H

= ˜ C

then goto LAB;

for i = 1 to n do

if i = 1 then [r(c) := α; L := L ∪ {c}; r(v

) := n · r

(L);

L := L ∪ {v

}];

else [r(v

) := α; L := L ∪ {v

}];

for j = 3 to d

− 1 do

r(v

) := 10 · r

(L); L := L ∪ {v

};

end for;

α := r

(e

);

end for;

LAB: r(˜ v) := α;

stop.

From the algorithm it follows immediately that |e

| ≥ 3 is needed and, furthermore, in case of ˜ H

= ˜ W

the spokes e

, . . . , e

must contain at least three vertices. We call all those vertices ˆ v of ˜ H

with r(ˆ v) := α in the algorithm α-vertices and the others β-vertices. Let A and B denote the set of α-vertices and β-vertices, respectively. Corresponding to the order of labelling we use the notation

v

= v

, v

= v

, v

, v

, . . . , v

= ˜ v

with q =

 

 

 



1 + ^P

i=1

(d

− 1), if H ˜

= ˜ C

, (n + 2) + ^P

i=1

(d

− 2), if H ˜

= ˜ W

. (4)

Concluding this section we give some useful properties of the labelling r needed in the following.

Lemma 1. Assume that the vertices of ˜ H

are labelled by the algorithm and consider the sequence v

, v

, . . . , v

. Then

∀ v

∈ B\{v

} : r(v

) > r

({v

, . . . , v

}), (5)

∀ v

∈ {c} ∪ V

\{v

, v

} : r(v

) > r(v

) + r(v

).

(6)

P roof. Condition (5) and condition (6) for β-vertices v

follow from the algorithm immediately. From (3) it follows that (6) is also true for α-vertices v

.

Lemma 2. Let M ⊆ V

, t ∈ {1, . . . , n} and a vertex subset M

⊆

^S

i=1

e

be arbitrarily chosen. Then

r

(M ) = r(v

) + r(v

) + r

(M

) ⇒ {v

, v

} ⊆ M.

P roof. Suppose v

∈ M . Then t ≥ 2 and we have to substitute r(v /

) = r

(e

) in the equation by using labels of other vertices. By Lemma 1, (5) this yields B

:= {v

, . . . , v

} ⊆ M , hence

r

(M \B

) = r(v

) + r(v

) + r(v

) + r

(M

).

Now observe that v

= v

∈ B

; furthermore, the condition (3) implies that v

is a β-vertex. By the first observation r(v

) has to be substituted by using labels of other vertices and the second observation implies (by Lemma 1, (5)) that this is not possible. Hence v

∈ M is necessary. Now the equation in Lemma 2 can be written as

r

(M \{v

}) = r(v

) + r

(M

)

and it follows immediately by Lemma 1, (5) that also the β-vertex v

must be contained in M .

3 The Sum Number of Hypercycles and Hyperwheels

In the following we use the labelling r and determine σ(C

) and σ(W

).

Observe that in every sum hypergraph the vertex with highest label is an isolated vertex.

Theorem 3. Let C

be a hypercycle with d(C

) ≥ 3. Then σ(C

) = 1.

(7)

P roof. We denote the vertices and edges of C

= (V

, E

) according to (1), take an isolated vertex ˜ v / ∈ V

and assume that the vertices of V

∪ {˜ v} are labelled by the algorithm given in the previous section. For convenience we denote by

B

= e

\{v

, v

}, if 1 ≤ t ≤ n − 1, B

= e

\{v

, v

, v

= v

}

(8)

subsets of β-vertices. Thus — considering B

, . . . , B

as vertex sequences

— (4) can be written into

v

, v

, B

, v

, B

, v

, B

, . . . , v

, B

, v

:= ˜ v,

where v

is the t-th α-vertex for t = 1, . . . , n + 1. Using r(v

) = r(v

) = 1 this yields

r(v

) = r(v

) + r(v

) + r

(B

) + x with x =

( 0, if 1 ≤ t ≤ n − 1, 1, if t = n.

(9)

Let E

denote the set of edges generated by the labelling r and the sum hypergraph property. Then the algorithm yields E

⊆ E

; we prove (7) by showing that even equality is fulfilled, i.e.,

∀ v

∈ V

∪ {˜ v} ∀ M ⊆ {v

, . . . , v

} : r(v

) = r

(M ) ∧ d(C

) ≥ 3 ⇒ M ∈ E

. (10)

By Lemma 1, (5) the condition r(v

) = r

(M ) can only be true if v

is an α-vertex v

with 1 ≤ t ≤ n. Hence by (9)

r

(M ) = r(v

) = r(v

) + r(v

) + r

(B

) + x.

Using (8) and Lemma 1, (5) we obtain B

⊆ M . Thus r

(M \B

) = r(v

) + r(v

) + x.

(11)

By Lemma 2 it follows that {v

, v

} ⊆ M and (11) can be written into r

(M \B

\{v

, v

}) = x.

(12)

In case of 1 ≤ t ≤ n − 1 (i.e., x = 0) this yields M = B

∪ {v

, v

} = e

. For t = n (i.e., x = 1) the only possibility is M = B

∪ {v

, v

, v

} = e

. Remark. The labelling r from the algorithm cannot be used to show σ(C

) = 1 if there is an edge e

, t − 1 ∈ {2, . . . , n − 1} with |e

| = 2.

(This is true because v

= v

is an α-vertex in this case and therefore r(v

) = r

(M ) = r

(e

\{v

}) + r

(e

), but e

∪ e

\{v

} / ∈ E

).

Now we turn to hyperwheels W

and start with two properties needed for the proof of the main result.

Lemma 4. Suppose that the hyperwheel W

= (V

, E

) with d(W

) ≥ 3 is denoted according to (2). Further let M ⊆ V

with |M | ≥ 2. If t ∈ {1, . . . , n}, r

(M ) = r(v

) and v

:= ˜ v, then

(i)

µ := |M ∩ {v

, . . . , v

}| = 1.

(13)

(ii) If j ∈ {1, . . . , t} with v

∈ M , then

M ∩

j−1

[

i=1

(e

\{v

, c}) = ∅.

(14)

P roof. (i) We use the notation (4) and consider the maximum index k in that sequence such that v

is an α-vertex belonging to M :

k = max{j ∈ {1, . . . , q} : v

∈ A ∧ v

∈ M }.

Then the algorithm yields that each β-vertex v

with m > k belongs to M . If M ∩ {v

, . . . , v

} = ∅, it follows that v

is such a β-vertex v

. Hence v

∈ M , i.e., µ = 1 in this case.

If M ∩{v

, . . . , v

} 6= ∅, suppose that j ∈ {2, . . . , t} is the maximum value such that v

∈ M . Then we obtain by Lemma 1, (5)

M

=



 [

i=j

(e

\{v

= c, v

, v

= v

})



 ⊆ M.

(15)

Note that M

= ∅ is possible. Now assume µ ≥ 2, i.e., there are k, j ∈ {1, . . . , t}, k < j such that {v

, v

} ⊆ M . With (15) and r

(M ) = r(v

) follows

r

(M \M

\{v

}) = (t − j + 1) · r(c) + r



 [

i=j

{v

}



 . (16)

Using j ≥ 2, t ≤ n and r(v

) = n · r

(V

∪ {c}) (cf., the algorithm) we obtain

r

(M \M

\{v

}) ≤ (n − 1) · r(c) + r

Ã

[

i=2

{v

}

! (17)

< r(v

) ≤ r(v

).

This is a contradiction to the assumption v

∈ M and therefore (13) is true.

(ii) Property (14) follows immediately from (17) because v

is the vertex with minimum label in

^S

i=1

(e

\{v

, c}).

Theorem 5. Let W

be a hyperwheel with d(W

) ≥ 3 and d(W

) <

2d(W

) − 1. Then

σ(W

) = 1.

(18)

P roof. 1. We denote the vertices and edges of W

= (V

, E

) according to (2), take an isolated vertex ˜ v / ∈ V

and assume that the vertices of V

∪ {˜ v}

are labelled by the algorithm given in the previous section. Further we denote with

B

= e

\{v

= c, v

, v

= v

}, 1 ≤ t ≤ n (19)

subsets of β-vertices. Using (8) and (19) — again considering B

, . . . , B

as vertex sequences — we can write (4) as

v

, v

, B

, v

, B

, . . . , v

, B

, v

= c, v

, B

, v

, B

, . . . , v

, B

, v

:= ˜ v,

where v

(for s 6= n + 1) and v

(for s = n + 1) is the s-th α-vertex for s = 1, . . . , 2n + 1.

2. Let E

denote the set of edges generated by the labelling r and the sum hypergraph property. Obviously, the algorithm yields E

⊆ E

; we prove (18) by showing that even equality is given, i.e.,

∀ v

∈ V

∪ {˜ v} ∀ M ⊆ {v

, . . . , v

} :

r(v

) = r

(M ) ∧ d(W

) ≥ 3 ∧ |M | ≤ d < 2d − 1 ⇒ M ∈ E

. (20)

In the proof of Theorem 3 the validity of (20) is shown for all vertices v

of the rim and the centre (because r(c) = r

(e

)). Hence it suffices to consider only vertices of V

\V

\{c} in the following. Because v

is a β-vertex the condition r(v

) = r

(M ) in (20) implies that v

is an α-vertex v

with 1 ≤ t ≤ n; hence

r

(M ) = r(v

) = r

(e

) = r(c) + r(v

) + r

(B

) + r(v

).

B

⊆ M follows from Lemma 1 (5) and we obtain r

(M \B

) = r(c) + r(v

) + r(v

).

(21)

It remains to show that (21) together with the condition |M | ≤ d < 2d − 1 from (20) implies M = e

.

3. We show that v

∈ M is necessary. By Lemma 4, (13) M contains

exactly one vertex v

∈ {v

, . . . , v

}; now assume j < t. Then t ≥ 2

and Lemma 1, (5) yields B

∪ B

⊆ M . We obtain with (19)

|M \B

\B

\{v

}| < (2d − 1) − 2(d − 3) − 1 = 4.

(22)

On the other hand, by (21) and r(v

) = r

(e

) it follows that r

(M \B

\B

\{v

}) > 2r(c).

Due to Lemma 4, (14) only (pairwise distinct) vertices of V

∪ {c} are avail- able to provide this value. By Lemma 1, (6) we need at least three such vertices to substitute r(c) once, i.e., (22) cannot be true. Hence v

∈ M and (21) can be written into

r

(M \B

\{v

}) = r(c) + r(v

).

(23)

4. Next we show that c ∈ M must be fulfilled. Assuming the contrary it follows with r(c) = r

(e

) that B

⊆ M . Hence by (8)

r

(M \B

\{v

}\B

) = r(v

) + r(v

) + r(v

) + r(v

).

Because of v

= v

Lemma 2 yields {v

, v

} ⊆ M , i.e., r

(M \B

\B

\{v

, v

, v

}) = r(v

) + 1.

Now observe that there is no vertex ˆ v with r(ˆ v) = r(v

) + 1 but this is a contradiction to

|M \B

\B

\{v

, v

, v

}| < (2d − 1) − 2(d − 3) − 3 = 2.

Thus c ∈ M and (23) can be written as

r

(M \B

\{v

, c}) = r(v

).

Because v

is a β-vertex we obtain by Lemma 1, (5) that r(v

) cannot be replaced by a sum of other labels, i.e., v

∈ M . Summarizing the results we have M = B

∪ {v

, c, v

} = e

and the proof is completed.

Remark. The labelling r from the algorithm can only be used to show σ(W

) = 1 if the condition d(W

) < 2d(W

) − 1 is true. (Otherwise observe that for some t ∈ {2, . . . , n−1} holds r(v

) = r

(e

) = r

(e

\{c})+

r

(e

); but e

∪ e

\{c} / ∈ E

and |e

∪ e

\{c}| ≥ 2d − 1).

Acknowledgement

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

References

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Math. Graph Theory 19 (1999) 79–91.

Received 4 May 1999

Revised 28 September 1999