Applied Mathematical Sciences, Vol. 9, 2015, no. 5, 243 - 253
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2015.411972
On the Set of Simple Hypergraph Degree
Sequences
Hasmik Sahakyan
Institute for Informatics and Automation Problems
National Academy of Sciences
1 P. Sevak st., 0014 Yerevan, Armenia
Copyright © 2014 Hasmik Sahakyan. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
For a given 𝑚, 0 < 𝑚 ≤ 2𝑛 , let 𝐷𝑚 (𝑛) denote the set of all hypergraphic
sequences for hypergraphs with 𝑛 vertices and 𝑚 hyperedges. A hypergraphic
sequence in 𝐷𝑚 (𝑛) is upper hypergraphic if all its components are at least 𝑚/2.
̂𝑚 (𝑛) denote the set of all upper hypergraphic sequences. A structural
Let 𝐷
̂𝑚 (𝑛) was
characterization of the lowest and highest rank maximal elements of 𝐷
provided in an earlier study. In the current paper we present an analogous
characterization for all upper non-hypergraphic sequences. This allows
determining the thresholds 𝑟̅𝑚𝑖𝑛 and 𝑟𝑚𝑎𝑥 such that all upper sequences of ranks
lower than 𝑟̅𝑚𝑖𝑛 are hypergraphic and all sequences of ranks higher than 𝑟𝑚𝑎𝑥 are
non-hypergraphic.
Keywords: hypergraph, degree sequence, complement
1. Introduction
A hypergraph 𝐻 is a pair (𝑉, 𝐸), where 𝑉 is the vertex set of 𝐻, and 𝐸, the set of
hyperedges, is a collection of non-empty subsets of 𝑉. The degree of a vertex 𝑣 of
𝐻, denoted by 𝑑(𝑣), is the number of hyperedges in 𝐻 containing 𝑣. A
hypergraph 𝐻 is simple if it has no repeated hyperedges. A hypergraph 𝐻 is 𝑟uniform if all hyperedges contain 𝑟-vertices.
Let 𝑉 = {𝑣1 , ⋯ , 𝑣𝑛 }. 𝑑(𝐻) = (𝑑(𝑣1 ), ⋯ , 𝑑(𝑣𝑛 )) is the degree sequence of
hypergraph 𝐻. A sequence 𝑑 = (𝑑1 , ⋯ , 𝑑𝑛 ) is hypergraphic if there is a simple
hypergraph 𝐻 with degree sequence 𝑑. For a given 𝑚, 0 < 𝑚 ≤ 2𝑛 , let 𝐻𝑚 (𝑛)
denote the set of all simple hypergraphs ([𝑛], 𝐸), where [𝑛] = {1,2, ⋯ , 𝑛}, and
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Hasmik Sahakyan
|𝐸| = 𝑚. Let 𝐷𝑚 (𝑛) denote the set of all hypergraphic sequences of hypergraphs
in 𝐻𝑚 (𝑛). The subject of our investigation is the set 𝐷𝑚 (𝑛), as well as its
complement, the set of integer 𝑛-tuples which are not hypergraphic sequences for
𝐻𝑚 (𝑛).
The problem of characterization of 𝐷𝑚 (𝑛) remains open even for 3-uniform
hypergraphs (see [4]-[13]). The problem has its interpretation in terms of
multidimensional binary cubes that arises out of the discrete isoperimetric
problem for n-dimensional binary cube [1-3]. In [6] the polytope of degree
sequences of uniform hypergraphs was studied and several partial results were
obtained. It was shown in [10] that any two 3-uniform hypergraphs can be
transformed into each other by using a sequence of trades. Several necessary and
one sufficient conditions were obtained for existence of simple 3-uniform
hypergraphs in [8]. Steepest degree sequences were defined in [7] and it was
shown that the whole set of degree sequences of simple uniform hypergraphs can
be determined by its steepest elements. Upper and lower degree sequences were
defined for 𝐷𝑚 (𝑛) in [11] where it was proven that the whole set 𝐷𝑚 (𝑛) can be
easily determined by the set of its upper and/or lower degree sequences. Upper
degree sequences of the lowest and highest ranks were characterized in our earlier
study [12]. In the current paper we extend the study to the complementary area of
𝐷𝑚 (𝑛), which can be supportive in solving the problem algorithmically.
𝑛
𝑛
Define the grid 𝛯𝑚+1
as: 𝛯𝑚+1
= {(𝑎1 , ⋯ , 𝑎𝑛 )|0 ≤ 𝑎𝑖 ≤ 𝑚 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖}, and place a
𝑛
component-wise partial order on 𝛯𝑚+1
: (𝑎1 , ⋯ , 𝑎𝑛 ) ≤ (𝑏1 , ⋯ , 𝑏𝑛 ) if and only if
𝑎𝑖 ≤ 𝑏𝑖 for all 𝑖. The rank 𝑟(𝑎1 , ⋯ , 𝑎𝑛 ) of an element (𝑎1 , ⋯ , 𝑎𝑛 ) is defined as
𝑛
(𝑎1 + ⋯ + 𝑎𝑛 ). The Hasse diagram of 𝛯𝑚+1
has 𝑚 ∙ 𝑛 + 1 levels according to the
ranks of elements: the 𝑖-th level contains all elements of the rank 𝑖.
𝑟(𝑎1 , ⋯ , 𝑎𝑛 ) = 𝑟(𝑏1 , ⋯ , 𝑏𝑛 ) + 1 if (𝑎1 , ⋯ , 𝑎𝑛 ) covers (𝑏1 , ⋯ , 𝑏𝑛 ) (see [14] for
𝑛
undefined terms). In this manner, 𝐷𝑚 (𝑛) is a subset of 𝛯𝑚+1
.
A hypergraphic sequence 𝑑 = (𝑑1 , ⋯ , 𝑑𝑛 ) ∈ 𝐷𝑚 (𝑛),
is called upper
hypergraphic if 𝑑𝑖 ≥ 𝑚𝑚𝑖𝑑 for all 𝑖, where 𝑚𝑚𝑖𝑑 = (𝑚 + 1)/2 for odd 𝑚 and
̂𝑚 (𝑛) denote the set of all upper hypergraphic
𝑚𝑚𝑖𝑑 = 𝑚/2 for even 𝑚. Let 𝐷
sequences in 𝐷𝑚 (𝑛). According to [11], for constructing all elements of 𝐷𝑚 (𝑛), it
̂𝑚 (𝑛), reducing in this manner the problem of
is sufficient to find elements of 𝐷
𝑛
̂ , where 𝐻
̂=
describing the set of degree sequences from 𝛯𝑚+1
to 𝐻
{(𝑎1 , ⋯ , 𝑎𝑛 )|𝑚𝑚𝑖𝑑 ≤ 𝑎𝑖 ≤ 𝑚 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖}.
̂𝑚 (𝑛) = 𝐷𝑚 (𝑛) ⋂ 𝐻
̂ . Figure 1 illustrates Hasse diagram of 𝛯53 .
Thus 𝐷
̂𝑚 (𝑛) and 𝐷𝑚 (𝑛) in 𝛯53 .
Figure 2 demonstrates 𝐷
̂ \𝐷
̂𝑚 (𝑛) is the set of all upper non-hypergraphic sequences. Recall
𝐹̂𝑚 (𝑛) = 𝐻
̂𝑚 (𝑛) is an ideal in 𝐻
̂ (𝐹̂𝑚 (𝑛) is a filter in 𝐻
̂ ).
([11]) that 𝐷
̂𝑚 (𝑛) and 𝐹̂𝑚 (𝑛) in 𝐻
̂.
Figure 3 illustrates 𝐷
On the set of simple hypergraph degree sequences
245
444
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432
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431
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030
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010
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021
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042
012
011
020
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034
024
033
023
013
014
004
003
002
001
000
Figure 1.
̂.
Circles correspond to elements/vertices of 𝛯53 . Highlighted part (gray) composes 𝐻
444
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441
430
421
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432
431
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422
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130
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101
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030
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132
112
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142
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031
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133
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042
032
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012
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143
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021
011
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041
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102
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040
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304
020
010
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110
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140
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333
323
303
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343
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211
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331
434
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104
022
013
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043
033
024
023
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004
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Figure 2.
034
̂𝑚 (𝑛).
Whole highlighted part composes 𝐷𝑚 (𝑛), and its lighter part is 𝐷
246
Hasmik Sahakyan
444
433
442
441
440
431
420
410
402
320
401
412
340
330
411
400
422
431
430
310
423
432
331
321
311
301
300
230
403
240
210
211
200
404
231
130
110
313
212
202
121
111
101
040
030
213
132
112
102
224
204
041
031
113
144
133
124
042
032
114
104
022
012
134
143
123
103
021
011
214
142
122
203
020
010
223
304
234
233
314
141
131
243
324
242
232
244
334
333
222
140
100
343
323
303
120
201
414
241
221
302
220
332
322
312
424
342
413
341
344
434
443
013
044
034
043
033
024
023
014
004
003
002
001
000
Figure 3.
̂ . Light part in 𝐻
̂ is 𝐷
̂𝑚 (𝑛), and dark part is 𝐹̂𝑚 (𝑛).
Highlighted part composes 𝐻
In [12] we obtained simple formulas for the lowest 𝑟𝑚𝑖𝑛 and the highest 𝑟𝑚𝑎𝑥
̂𝑚 (𝑛).
ranks of maximal elements in 𝐷
In this paper we present analogous results for non-hypergraphic sequences,
namely we seek for the lowest and highest ranks 𝑟̅𝑚𝑖𝑛 and 𝑟̅𝑚𝑎𝑥 , respectively, of
minimal elements of 𝐹̂𝑚 (𝑛). Section 2 determines a characterization of the lowest
rank. We obtain a series of minimal elements of 𝐹̂𝑚 (𝑛) and prove that these
elements are the lowest rank non-hypergraphic sequences. Section 3 determines
̂ of ranks
the highest rank minimal elements. We conclude that all sequences in 𝐻
̂ of ranks higher than 𝑟𝑚𝑎𝑥
lower than 𝑟̅𝑚𝑖𝑛 are hypergraphic and all sequences in 𝐻
are non-hypergraphic. In the last section we give some estimates on lowest and
̂ depending on the values of 𝑚.
highest ranks in 𝐻
2. Lowest rank
In this section we provide a characterization of the lowest rank minimal elements
of 𝐹̂𝑚 (𝑛). We obtain a series of minimal elements of 𝐹̂𝑚 (𝑛) and prove that these
elements are the lowest rank non-hypergraphic sequences.
Let 𝑚 be given in the standard binary representation form:
𝑚 = 2𝑘1 + ⋯ + 2𝑘𝑝 where 𝑘1 > ⋯ > 𝑘𝑝 > 0.
(1)
̂𝑚 (𝑛) is defined in [12] as follows:
The lowest rank 𝑟𝑚𝑖𝑛 of maximal elements of 𝐷
𝑝
𝑘
𝑘
𝑟𝑚𝑖𝑛 = ∑𝑖=1((𝑛 − 𝑘𝑖 − (𝑖 − 1)) ∙ 2 𝑖 + 𝑘𝑖 ∙ 2 𝑖 −1 ).
On the set of simple hypergraph degree sequences
247
̂𝑚 (𝑛) corresponds to a hypergraph
The maximal element 𝑑𝑚𝑖𝑛 of rank 𝑟𝑚𝑖𝑛 in 𝐷
whose edges are identified with the initial 𝑚-segment of the reverse lexicographic
ordering of [2𝑛 ] and is unique (up to coordinate permutations).
𝑑𝑖
𝑑𝑖
𝑑𝑖
𝑑𝑖
𝑗−1
= (∑𝑙=1 2𝑘𝑙−1 ) + 2𝑘𝑗 for 𝑖 = 𝑘𝑗 + 1, 𝑗 = 1, ⋯ , 𝑝,
𝑗
= (∑𝑙=1 2𝑘𝑙−1 ) + (∑𝑝𝑙=𝑗+1 2𝑘𝑗 ) for 𝑘𝑗+1 + 2 ≤ 𝑖 ≤ 𝑘𝑗 , 𝑗 = 1, ⋯ , 𝑝 − 1,
= (∑𝑝𝑙=1 2𝑘𝑙−1 ) = 𝑚/2 for 1 ≤ 𝑖 ≤ 𝑘𝑝 ,
𝑝
= (∑𝑙=1 2𝑘𝑙 ) = 𝑚 for 𝑘1 + 2 ≤ 𝑖 ≤ 𝑛.
Thus 𝑑𝑚𝑖𝑛 has the following form:
𝑛2
𝑛1
𝑑𝑚𝑖𝑛
𝑛3
⏞ ⋯,𝑚,⏞
= (𝑚,
𝑑𝑛1 +1 , ⋯ , 𝑑𝑛1 +𝑛2 , ⏞
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 )
(2)
where 𝑚 > 𝑑𝑛1 +1 ≥ ⋯ ≥ 𝑑𝑛1 +𝑛2 > 𝑚𝑚𝑖𝑑 ;
𝑛1 + 𝑛2 + 𝑛3 = 𝑛. Notice that
𝑘1
𝑑𝑛1 +1 = 2 .
Below, we establish two easily verified properties of 𝑑𝑚𝑖𝑛 which will be used to
prove our results.
Property 1. 𝑑𝑖 is the largest possible value for fixed 𝑑1 , ⋯ , 𝑑𝑖−1.
Property 2.
a) 𝑛1 > 0 if and only if 𝑚 ≤ 2𝑛−1. If 2𝑡−1 < 𝑚 ≤ 2𝑡 for some 𝑡 ≤ 𝑛 then
𝑛1 = 𝑛 − 𝑡.
b) 𝑛2 = 0 if and only if 𝑚 = 2𝑡 for some 𝑡.
c) 𝑛3 = 𝑘𝑝 .
We will also use the notions of flatter and steeper elements defined as follows:
Let 𝑎𝑖 ≥ 𝑎𝑗 + 2 for some 1 ≤ 𝑖, 𝑗 ≤ 𝑛, then (𝑎1 , ⋯ , 𝑎𝑖 − 1, ⋯ , 𝑎𝑗 + 1, ⋯ , 𝑎𝑛 ) is
flatter than (𝑎1 , ⋯ , 𝑎𝑛 ) and (𝑎1 , ⋯ , 𝑎𝑛 ) is steeper than (𝑎1 , ⋯ , 𝑎𝑖 − 1, ⋯ , 𝑎𝑗 +
1, ⋯ , 𝑎𝑛 ). If (𝑎1 , ⋯ , 𝑎𝑛 ) ∈ 𝐷𝑚 (𝑛), then all elements flatter than (𝑎1 , ⋯ , 𝑎𝑛 ) also
belong to 𝐷𝑚 (𝑛) (see [7]).
The following theorem determines a minimal element of the lowest rank of
𝐹̂𝑚 (𝑛).
Theorem 1. Let 𝑑𝑚𝑖𝑛 be presented as in (2).
(1) If 𝑚 ≠ 2𝑡 for arbitrary 𝑡, then:
𝑛1
𝑛−𝑛1 −1
⏞ ⋯ , 𝑚 , 2𝑘1 + 1, ⏞
a) 𝑑̅𝑚𝑖𝑛 = (𝑚,
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 )
𝐹̂𝑚 (𝑛), where 2𝑘1 is the first component in (1).
b) 𝑟(𝑑̅𝑚𝑖𝑛 ) = 𝑟̅𝑚𝑖𝑛 .
(2) If 𝑚 = 2𝑡 for some 𝑡, then:
is a minimal element of
248
Hasmik Sahakyan
𝑛1
𝑛−𝑛1 −1
⏞ ⋯ , 𝑚 , 𝑚𝑚𝑖𝑑 + 1, ⏞
a) 𝑑̅𝑚𝑖𝑛 = (𝑚,
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ) is a minimal element of
𝐹̂𝑚 (𝑛).
b) 𝑟(𝑑̅𝑚𝑖𝑛 ) = 𝑟̅𝑚𝑖𝑛 .
Proof. We consider both cases separately.
(1) 𝑚 ≠ 2𝑡 . Then 𝑛2 > 0 by Property 2.
̂ and all
a) Here it suffices to show that 𝑑̅𝑚𝑖𝑛 is a non-hypergraphic sequence in 𝐻
̅
̅
̂ covered by 𝑑𝑚𝑖𝑛 are hypergraphic in 𝐻
̂ . 𝑑𝑚𝑖𝑛 ∉ 𝐷
̂𝑚 (𝑛) by Property
elements of 𝐻
̅
̂
1. All elements of 𝐻 covered by 𝑑𝑚𝑖𝑛 have one of the following forms:
𝑛1
𝑛−𝑛1 −1
⏞ ⋯ , 𝑚 , 2 𝑘1 , ⏞
(𝑚,
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 )
𝑛1
(3)
𝑛−𝑛1 −1
⏞ ⋯ , 𝑚, 𝑚 − 1 , 2𝑘1 + 1, ⏞
(𝑚,
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 )
(4)
̂𝑚 (𝑛). If 𝑚 = 2𝑘1 + 1, then
Sequence (3) is less than 𝑑𝑚𝑖𝑛 and hence belongs to 𝐷
𝑘1
sequence (4) is just a permutation of (3). If 𝑚 ≥ 2 + 2, then (4) is flatter than
(3), and therefore is hypergraphic.
̂ of 𝑟(𝑎) < 𝑟(𝑑̅𝑚𝑖𝑛 ) belong to 𝐷
̂𝑚 (𝑛). It is
b) We have to prove that all 𝑎 ∈ 𝐻
̂ of 𝑟(𝑎) = 𝑟(𝑑̅𝑚𝑖𝑛 ) − 1 belong to 𝐷
̂𝑚 (𝑛).
sufficient to show that all 𝑎 ∈ 𝐻
𝑛1
𝑛−𝑛1 −1
⏞ ⋯,𝑚,2 ,⏞
Consider 𝑑 = (𝑚,
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ) which is of rank 𝑟(𝑑̅𝑚𝑖𝑛 ) − 1.
′
𝑘1
̂ of the rank 𝑟(𝑑̅𝑚𝑖𝑛 ) − 1 can be obtained from 𝑑 ′ by
All elements in 𝐻
combinations of the following unit operations:
i) replace (𝑚, 2𝑘1 ) by (𝑚 − 1,2𝑘1 + 1);
ii) replace (𝑚, 𝑚𝑚𝑖𝑑 ) by (𝑚 − 1, 𝑚𝑚𝑖𝑑 + 1);
iii) replace (2𝑘1 , 𝑚𝑚𝑖𝑑 ) by (2𝑘1 − 1, 𝑚𝑚𝑖𝑑 + 1).
In all three cases above the resulting sequence is either flatter than 𝑑 ′ or is a
̂𝑚 (𝑛).
permutation of 𝑑′ , and therefore belongs to 𝐷
𝑡
(2) 𝑚 = 2 . Then 𝑛2 = 0 by Property 2.
̂𝑚 (𝑛) and all elements of 𝐻
̂
a) Here it is only required to prove that 𝑑̅𝑚𝑖𝑛 ∉ 𝐷
̂ . 𝑑̅𝑚𝑖𝑛 ∉ 𝐷
̂𝑚 (𝑛) because
covered by 𝑑̅𝑚𝑖𝑛 are hypergraphic sequences in 𝐻
̅
̂𝑚 (𝑛).
𝑑𝑚𝑖𝑛 > 𝑑𝑚𝑖𝑛 and 𝑑𝑚𝑖𝑛 is a maximal element of 𝐷
̅
̂
All elements of 𝐻 covered by 𝑑𝑚𝑖𝑛 have one of the following forms:
𝑛3
𝑛1
⏞ ⋯,𝑚,⏞
(𝑚,
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ) or
𝑛1
𝑛3 −1
⏞ ⋯ , 𝑚, 𝑚 − 1 , 𝑚𝑚𝑖𝑑 + 1, ⏞
(𝑚,
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ).
On the set of simple hypergraph degree sequences
249
In the former form this is just 𝑑𝑚𝑖𝑛 and in the latter form this is flatter than
̂𝑚 (𝑛).
𝑑𝑚𝑖𝑛 , and hence belongs to 𝐷
̂ belong
b) We have to prove that all elements 𝑎 of 𝑟(𝑎) = 𝑑̅𝑚𝑖𝑛 − 1 in 𝐻
𝑛1
𝑛3
̂𝑚 (𝑛). These are flatter than 𝑑𝑚𝑖𝑛 = (𝑚,
⏞ ⋯,𝑚,⏞
to 𝐷
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ), and
̂𝑚 (𝑛).
hence belong to 𝐷

Remark that in case of hypergraphic sequences there is a unique maximal element
̂𝑚 (𝑛), whereas in case of non-hypergraphic sequences there are a
of rank 𝑟𝑚𝑖𝑛 in 𝐷
number of minimal elements of rank 𝑟̅𝑚𝑖𝑛 in 𝐹̂𝑚 (𝑛).
Theorem 2 below produces a series of minimal elements of 𝐹̂𝑚 (𝑛).
Theorem 2.
(1) If 𝑚 ≠ 2𝑡 then
𝑛1
𝑛−𝑛1 −1
⏞ ⋯ , 𝑚, 𝑚 − 𝑡 , 2𝑘1 + 𝑡 + 1, ⏞
(𝑚,
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ), 𝑡 = 1, ⋯ , (𝑚 − 2𝑘1 − 1)/2,
are minimal elements of 𝑟̅𝑚𝑖𝑛 in 𝐹̂𝑚 (𝑛).
(2) If 𝑚 = 2𝑡 then
𝑛1
𝑛−𝑛1 −1
⏞ ⋯ , 𝑚, 𝑚 − 𝑡 , 𝑚𝑚𝑖𝑑 + 𝑡 + 1, ⏞
(𝑚,
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ) 𝑡 = 1, ⋯ , (𝑚 − 𝑚𝑚𝑖𝑑 − 1)/2,
are minimal elements of 𝑟̅𝑚𝑖𝑛 in 𝐹̂𝑚 (𝑛).
The proof is obtained by an analogous reasoning as in Theorem 1.
3. Highest rank
In this section we determine a characterization of highest rank minimal elements
of 𝐹̂𝑚 (𝑛).
Let 𝑚 be given in the following canonical representation form:
𝑚 = 𝐶𝑛𝑛 + 𝐶𝑛𝑛−1 + ⋯ + 𝐶𝑛𝑛−𝑘 + 𝑚1 , 𝑚1 < 𝐶𝑛𝑛−𝑘−1
(5)
̂𝑚 (𝑛), be defined as in [12]:
Let the highest rank 𝑟𝑚𝑎𝑥 of maximal elements of 𝐷
𝑟𝑚𝑎𝑥 = ∑𝑘𝑖=0((𝑛 − 𝑖) ∙ 𝐶𝑛𝑛−𝑖 ) + (𝑛 − 𝑘 − 1) ∙ 𝑚1 .
̂𝑚 (𝑛) of the rank 𝑟𝑚𝑎𝑥 . 𝐷𝑚𝑎𝑥
Let 𝐷𝑚𝑎𝑥 denote the class of maximal elements of 𝐷
defines the set of degree sequences of that class of hypergraphs which have 𝐶𝑛𝑛 +
𝐶𝑛𝑛−1 + ⋯ + 𝐶𝑛𝑛−𝑘 common hyperedges (the subsets of [𝑛] of cardinalities 𝑛, 𝑛 −
1, ⋯ , 𝑛 − 𝑘) and differ only in the remaining 𝑚1 hyperedges (the (𝑛 − 𝑘 − 1)𝑚1
subsets of [𝑛]). Thus, |𝐷𝑚𝑎𝑥 | = 𝐶𝐶 𝑛−𝑘−1
. The components of all 𝑑𝑚𝑎𝑥 ∈ 𝐷𝑚𝑎𝑥
𝑛
250
Hasmik Sahakyan
𝑛−𝑖−1
are calculated as follows: 𝑑𝑖 = ∑𝑘𝑖=0 𝐶𝑛−1
+ 𝑠𝑖 , where (𝑠1 , ⋯ , 𝑠𝑛 ) defines the
set of hypergraphic sequences for (𝑛 − 𝑘 − 1)-uniform hypergraphs with 𝑚1
edges.
Theorem 3.
Let 𝑚 = 𝐶𝑛𝑛 + 𝐶𝑛𝑛−1 + ⋯ + 𝐶𝑛𝑛−𝑘 + 𝑚1 , 𝑚1 < 𝐶𝑛𝑛−𝑘−1.
𝑘
a) If 𝑚1 ≥ 1 + − 𝑘 − 1, then 𝑟̅𝑚𝑎𝑥 = 𝑟𝑚𝑎𝑥 + 1.
𝑛
𝑘
b) If 𝑚1 < 1 + 𝑛 − 𝑘 − 1, then 𝑟̅𝑚𝑎𝑥 ≤ 𝑟𝑚𝑎𝑥 .
Proof.
a) It is easy to check that there is a sequence 𝑑 ′ = (𝑑1 , ⋯ , 𝑑𝑛 ) in 𝐷𝑚𝑎𝑥 such that
𝑑1 ≥ ⋯ ≥ 𝑑𝑛−1 > 𝑑𝑛 (we can always choose 𝑚1 hyperedges such that 𝑠1 ≥ ⋯ ≥
𝑠𝑛−1 > 𝑠𝑛 ). Consider 𝑑̅ ′ = (𝑑1 , ⋯ , 𝑑𝑛−1 , 𝑑𝑛 + 1) , which belongs to 𝐹̂𝑚 (𝑛). All
elements covered by 𝑑̅ ′ are either flatter than 𝑑′ or permutations of 𝑑 ′ and, thus,
̂𝑚 (𝑛). Therefore 𝑑̅′ is a minimal element of the rank 𝑟𝑚𝑎𝑥 + 1 in
belong to 𝐷
̂ belong to 𝐹̂𝑚 (𝑛), there is no
𝐹̂𝑚 (𝑛). Since all elements of the rank 𝑟𝑚𝑎𝑥 + 1 in 𝐻
̂
minimal element of 𝐹𝑚 (𝑛) of rank higher than 𝑟𝑚𝑎𝑥 + 1. Thus 𝑟̅𝑚𝑎𝑥 = 𝑟𝑚𝑎𝑥 + 1.
b)
This case is obvious.
4. Concluding remarks
̂ depending
The last section gives estimates on the lowest and highest ranks in 𝐻
on parameter 𝑚 and brings several concluding remarks.
As it was stated above all sequences in 𝑚 with ranks lower than 𝑟̅𝑚𝑖𝑛 are
̂ with ranks higher than 𝑟𝑚𝑎𝑥 are nonhypergraphic and all sequences in 𝐻
̂𝑚 (𝑛) and all minimal elements of
hypergraphic. Hence all maximal elements of 𝐷
̂
𝐹𝑚 (𝑛) have ranks ranging between 𝑟̅𝑚𝑖𝑛 and 𝑟𝑚𝑎𝑥 + 1, and, thus, are located
̂ . An illustration of the upper hypergraphic
between 𝑟̅𝑚𝑖𝑛 and 𝑟𝑚𝑎𝑥 + 1 levels of 𝐻
̂ for 𝑛 = 3 and 𝑚 = 4 is given in Figure 3.
and non-hypergraphic sequences in 𝐻
̂𝑚 (𝑛) = {(3,3,3),
𝐷
(4,2,2), (3,3,2), (3,2,3), (2,4,2), (2,3,3), (2,2,4),
(3,2,2), (2,3,2), (2,2,3),
(2,2,2)}.
̂𝑚 (𝑛) are: (3,3,3), (4,2,2), (2,4,2), (2,2,4).
Maximal elements of 𝐷
𝐹̂𝑚 (𝑛) = {(4,4,4),
(4,4,3), (4,3,4), (3,4,4),
(4,4,2), (4,3,3), (4,2,4), (3,4,3), (3,3,4), (2,4,4),
(4,3,2), (4,2,3), (3,4,2), (3,2,4), (2,4,3), (2,3,4)}.
Maximal elements of 𝐹̂𝑚 (𝑛) are: (4,3,2), (4,2,3), (3,4,2), (3,2,4), (2,4,3), (2,3,4).
And thus 𝑟𝑚𝑖𝑛 = 8, 𝑟𝑚𝑎𝑥 = 9, 𝑟̅𝑚𝑖𝑛 = 9 and 𝑟̅𝑚𝑎𝑥 = 9.
On the set of simple hypergraph degree sequences
251
̂ : the lowest level consists
We consider the lowest, highest and middle levels in 𝐻
of the lowest element (𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ), the highest level consists of the highest
element (𝑚, ⋯ , 𝑚) and the middle level consists of all elements of the rank
𝑛 ∙ (𝑚 + 𝑚𝑚𝑖𝑑 )/2, particularly it contains the element ((𝑚 + 𝑚𝑚𝑖𝑑 )/2, ⋯ , (𝑚 +
𝑚𝑚𝑖𝑑 )/2).
Next we shall examine the distance of 𝑟̅𝑚𝑖𝑛 and 𝑟𝑚𝑎𝑥 ranks/levels from the lowest,
̂.
middle and highest levels of 𝐻
First distinguish the following cases for 𝑟̅𝑚𝑖𝑛 :
a) If 2𝑡−1 < 𝑚 < 2𝑡 for some 𝑡 ≤ 𝑛, then:
𝑚−𝑡
𝑡−1
⏞ ⋯ , 𝑚, 2𝑡−1 + 1, ⏞
𝑑̅𝑚𝑖𝑛 = (𝑚,
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 )
𝑟̅𝑚𝑖𝑛 = (𝑛 − 𝑡) ∙ 𝑚 + 2𝑡−1 + (𝑡 − 1) ∙ 𝑚𝑚𝑖𝑑 + 1
̂ will be:
The distance from the lowest level in 𝐻
𝑚
𝑚
𝑟̅𝑚𝑖𝑛 − 𝑟(𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ) ≈ (𝑛 − 𝑡) ∙ + (2𝑡−1 − + 1) =
2
2
𝑚
𝑡−1
∙ (𝑛 − 𝑡 − 1) + 2
+1
2
̂ will be:
The distance from the middle level in 𝐻
𝑚
𝑟̅𝑚𝑖𝑛 − 𝑟((𝑚 + 𝑚𝑚𝑖𝑑 )/2, ⋯ , (𝑚 + 𝑚𝑚𝑖𝑑 )/2) ≈ (𝑛 − 𝑡) ∙ 4 − (𝑡 − 1) ∙
𝑚
4
+ 2𝑡−1 + 1 − 3𝑚/4 =
𝑚
4
∙ (𝑛 − 2𝑡 − 2) + 2𝑡−1 + 1.
b) If 𝑚 = 2𝑡 for some 𝑡, then:
𝑚−𝑡
𝑡−1
⏞ ⋯ , 𝑚, 𝑚𝑚𝑖𝑑 + 1, ⏞
𝑑̅𝑚𝑖𝑛 = (𝑚,
𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 )
𝑟̅𝑚𝑖𝑛 = (𝑛 − 𝑡) ∙ 𝑚 + 𝑡 ∙ 𝑚𝑚𝑖𝑑 + 1
The distance from the lowest level will be:
𝑟̅𝑚𝑖𝑛 − 𝑟(𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ) ≈
𝑚
∙ (𝑛 − 𝑡) + 1
2
The distance from the middle level will be:
𝑟̅𝑚𝑖𝑛 − 𝑟((𝑚 + 𝑚𝑚𝑖𝑑 )/2, ⋯ , (𝑚 + 𝑚𝑚𝑖𝑑 )/2) ≈ (𝑛 − 𝑡) ∙
𝑚
4
∙ (𝑛 − 2𝑡) + 1.
𝑚
4
−𝑡∙
𝑚
4
+1=
̂ with
As we see in both cases 𝑟̅𝑚𝑖𝑛 goes up from the lowest level in 𝐻
decrease of 𝑚.
252
Hasmik Sahakyan
𝑛−𝑖−1
Now we estimate the case of 𝑟𝑚𝑎𝑥 . We have 𝑟𝑚𝑎𝑥 = 𝑛 ∙ ∑𝑘𝑖=0 𝐶𝑛−1
+ 𝑚1 ∙
(𝑛 − 𝑘 − 1).
𝑛−𝑖−1
𝑛−𝑘−1
a)
𝑚1 = 0. 𝑚 = ∑𝑘𝑖=0 𝐶𝑛𝑛−𝑖 = 2 ∙ ∑𝑘−1
+ 𝐶𝑛−1
, and thus 𝑚𝑚𝑖𝑑 =
𝑖=0 𝐶𝑛−1
𝑘−1 𝑛−𝑖−1
𝑛−𝑘−1
∑𝑖=0 𝐶𝑛−1 + 𝐶𝑛−1 /2.
On the other hand, all components of 𝑑𝑚𝑎𝑥 are:
𝑛−𝑖−1
𝑑𝑖 = ∑𝑘𝑖=0 𝐶𝑛−1
.
̂ element is:
The distance from the lowest level in 𝐻
𝑛−𝑘−1
𝑟𝑚𝑎𝑥 − 𝑟(𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ) = 𝐶𝑛−1 ∙ 𝑛/2.
The distance from the highest level is:
𝑛−𝑖−1
𝑛−𝑘−1
𝑟(𝑚, ⋯ , 𝑚) − 𝑟𝑚𝑎𝑥 = 𝑛 ∙ (2 ∙ ∑𝑘−1
+ 𝐶𝑛−1
)−
𝑖=0 𝐶𝑛−1
𝑘
𝑘−1 𝑛−𝑖−1
𝑛−𝑖−1
𝑛 ∙ ∑𝑖=0 𝐶𝑛−1 = 𝑛 ∙ ∑𝑖=0 𝐶𝑛−1 .
𝑛−𝑖−1
𝑛−𝑘−1
b)
𝑚1 > 0. 𝑚 = ∑𝑘𝑖=0 𝐶𝑛𝑛−𝑖 + 𝑚1 = 2 ∙ ∑𝑘−1
+ 𝐶𝑛−1
+ 𝑚1 , and
𝑖=0 𝐶𝑛−1
𝑘−1 𝑛−𝑖−1
𝑛−𝑘−1
thus 𝑚𝑚𝑖𝑑 = ∑𝑖=0 𝐶𝑛−1 + (𝐶𝑛−1 + 𝑚1 )/2.
The distance from the lowest level is:
𝑚
𝑛−𝑘−1 𝑛
𝑟𝑚𝑎𝑥 − 𝑟(𝑚𝑚𝑖𝑑 , ⋯ , 𝑚𝑚𝑖𝑑 ) = 𝐶𝑛−1
∙ 2 + 𝑚1 ∙∙ (𝑛 − 𝑘 − 1) − 𝑛 ∙ 21 =
𝑛
𝑛
𝑛−𝑘−1
𝐶𝑛−1
∙ 2 + 𝑚1 ( 2 − 𝑘 − 1).
The distance from the highest level is:
𝑛−𝑖−1
𝑛 ∙ ∑𝑘−1
− 𝑚1 ∙ (𝑛 − 𝑘 − 1).
𝑖=0 𝐶𝑛−1
̂ depending on
Thus we have determined the layouts of 𝑟𝑚𝑎𝑥 and 𝑟̅𝑚𝑖𝑛 in 𝐻
parameter 𝑚. The obtained formulas show when 𝑟𝑚𝑎𝑥 and 𝑟̅𝑚𝑖𝑛 are close, or when
𝑟̅𝑚𝑖𝑛 is greater than the middle rank depending on 𝑚, etc.
References
[1] L. A. Aslanyan, Isoperimetry problem and related extremal problems of
discrete spaces, Problemy Kibernet. 36 (1979) 85-126.
[2] L. Aslanyan, H. Danoyan, Complexity of Hash-Coding type search algorithms
with perfect codes, Journal of Next Generation Information Technology (JNIT),
Volume 5, Number 4, November 2014, 26-35.
[3] L. Aslanyan, H. Danoyan, On the optimality of the Hash-Coding type nearest
neighbour search algorithm”, Selected Revised Papers of 9th CSIT conference,
submited to IEEE Xplore, 2013.
http://dx.doi.org/10.1109/csitechnol.2013.6710336
[4] L. Aslanyan, H. Sahakyan, Numerical characterization of n-cube subset
partioning, Electronic Notes in Discrete Mathematics, Volume 27, Pages 3-4/10,
October 2006, Elsevier B.V.,ODSA 2006 -Conference on Optimal Discrete
Structures and algorithms. http://dx.doi.org/10.1016/j.endm.2006.08.027
On the set of simple hypergraph degree sequences
253
[5] C. Berge, Hypergraphs. North Holland, Amsterdam, 1989.
[6] N. L. Bhanu Murthy and Murali K. Srinivasan, The polytope of degree
sequences of hypergraphs. Linear algebra and its applications, 350 (2002), 147170. http://dx.doi.org/10.1016/s0024-3795(02)00272-0
[7] D. Billington, Lattices and Degree Sequences of Uniform Hypergraphs. Ars
Combinatoria, 21A, 1986, 9-19.
[8] D. Billington, Conditions for degree sequences to be realisable by 3-uniform
hypergraphs, 1988.
[9] C. J. Colbourn, W. L. Kocay, and D. R. Stinson, Some NP-complete problems
for hypergraph degree sequences, Discrete Applied Maths. 14 (1986), 239-254.
http://dx.doi.org/10.1016/0166-218x(86)90028-4
[10] W. Kocay, P. C. Li, On 3-hypergraphs with equal degree sequences, Ars
Combin. 82 (2007), 145-157.
[11] H. Sahakyan, Numerical characterization of n-cube subset partitioning,
Discrete Applied Mathematics 157 (2009) 2191-2197.
http://dx.doi.org/10.1016/j.dam.2008.11.003
[12] H. Sahakyan, Essential points of the n-cube subset partitioning
characterization, Discrete Applied Mathematics 163, Part 2 (2014) 205-213.
http://dx.doi.org/10.1016/j.dam.2013.07.015
[13] H. Sahakyan, (0, 1)-matrices with different rows, Selected Revised Papers of
9th CSIT conference, submited to IEEE Xplore, 2013.
http://dx.doi.org/10.1109/csitechnol.2013.6710342
[14] R. Steven, Lattices and Ordered Sets, 2008, ISBN 0-387-78900-2, 305.
http://dx.doi.org/10.1007/978-0-387-78901-9
Received: December 1, 2014; Published: January 3, 2015