Dimensions of Hypergraphs - Semantic Scholar
JOURNAL
OF COMBINATORIAL
THEORY,
Series B 56, 278-295 (1992)
Dimensions
of Hypergraphs
PETER C. FISHBURN AT&T
Bell Laboratories,
Murray
Hill,
New Jersey
07974
W. T. TROTTER Bellcore,
445 South Street, Morristown, New Jersey 07962 Arizona State University, Tempe, Arizona 85287 Communicated
and
by the Editors
Received December 20, 1989
The dimension D(S) of a family S of subsets of n = (1,2, .... n> is defined as the minimum number of permutations of n such that every A E S is an intersection of initial segments of the permutations. Equivalent characterizations of D(S) are given in terms of suitable arrangements, interval dimension, order dimension, and the chromatic number of an associated hypergraph. We also comment on the maximum-sized family of k-element subsets of n having dimension m, and on the dimension of the family of all k-element subsets of n. The paper concludes with a series of alternative characterizations of D(S) = 2 and a list of open problems. 0 1992 Academic
Press, Inc.
1. INTRODUCTION We define the dimension D of a finite hypergraph as the minimum number of permutations of its ground set such that every edge of the hypergraph is the intersection of initial segments of the permutations. The paper relates D to other notions of dimensionality and to chromatic numbers of certain graphs and hypergraphs, summarizes prior results that translate into facts about D, proves some new results, and identifies problems for further research. Because many of the theorems for D are not new, a main purpose of our study is to interpret and organize other topics under one elementary concept. Primary connections to the present definition of the dimension of a hypergraph are provided by the theory of k-suitable arrangements initiated in Dushnik [8], the interval dimension of height 1 partially ordered sets from ‘I’rotter and Moore [28] and Trotter [25], and the notion of biorder dimension and related work on chromatic 278 0095-8956192 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.
DIMENSIONS
OF HYPERGRAPHS
2’79
numbers in Bouchet [2], Cogis [S], and Doignon, Ducamp, and Falmagne [7]. The last of these, referred to henceforth as DDF, and West [30] are particularly rich sources of information on notions of dimensionality of ordered sets and their ties to chromatic numbers. Let n denote a positive integer. Unless we say otherwise, a finite hypergraph is viewed as a pair H = (n, S) in which n = (1,2, .... n > is the ground set and S is a family of subsets of n called edges. If every edge in S is an r-element subset of n then H is said to be r-uniform. A graph is a 2-uniform hypergraph. An arrangement or permutation of n is a linear array TV= a, a2 . . . a, of all n points in n. The initial segmentsof TVare 0, (al >, (a,, a2), .... n. A set R of permutations of n realizes H = (n, S) if for every nonempty edge A in S an initial segment s, can be chosen for each Q E R so that A=
f-j
S,.
A moment’s reflection shows that R realizes H if and only if, for every A E S and all x E n\A, x follows an initial segment s of some 0 E R for which A c s. Moreover, if A and B are in the family of subsets obtained as intersections of initial segments of R, then A n B is also in this family. Note that the trivial hypergraph (n, 0) is realized by the empty set of permutations. DEFINITION. The dimensionD(H) of H = (n, S) is the smallest m > 0 for which there exists a,set R of m permutations of n which realizes H. Clearly different at right elements.
D(n, 0) = 0, o(n, (0 >) = 1, and, by any n permutations with last elements, D
OF COMBINATORIAL
THEORY,
Series B 56, 278-295 (1992)
Dimensions
of Hypergraphs
PETER C. FISHBURN AT&T
Bell Laboratories,
Murray
Hill,
New Jersey
07974
W. T. TROTTER Bellcore,
445 South Street, Morristown, New Jersey 07962 Arizona State University, Tempe, Arizona 85287 Communicated
and
by the Editors
Received December 20, 1989
The dimension D(S) of a family S of subsets of n = (1,2, .... n> is defined as the minimum number of permutations of n such that every A E S is an intersection of initial segments of the permutations. Equivalent characterizations of D(S) are given in terms of suitable arrangements, interval dimension, order dimension, and the chromatic number of an associated hypergraph. We also comment on the maximum-sized family of k-element subsets of n having dimension m, and on the dimension of the family of all k-element subsets of n. The paper concludes with a series of alternative characterizations of D(S) = 2 and a list of open problems. 0 1992 Academic
Press, Inc.
1. INTRODUCTION We define the dimension D of a finite hypergraph as the minimum number of permutations of its ground set such that every edge of the hypergraph is the intersection of initial segments of the permutations. The paper relates D to other notions of dimensionality and to chromatic numbers of certain graphs and hypergraphs, summarizes prior results that translate into facts about D, proves some new results, and identifies problems for further research. Because many of the theorems for D are not new, a main purpose of our study is to interpret and organize other topics under one elementary concept. Primary connections to the present definition of the dimension of a hypergraph are provided by the theory of k-suitable arrangements initiated in Dushnik [8], the interval dimension of height 1 partially ordered sets from ‘I’rotter and Moore [28] and Trotter [25], and the notion of biorder dimension and related work on chromatic 278 0095-8956192 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.
DIMENSIONS
OF HYPERGRAPHS
2’79
numbers in Bouchet [2], Cogis [S], and Doignon, Ducamp, and Falmagne [7]. The last of these, referred to henceforth as DDF, and West [30] are particularly rich sources of information on notions of dimensionality of ordered sets and their ties to chromatic numbers. Let n denote a positive integer. Unless we say otherwise, a finite hypergraph is viewed as a pair H = (n, S) in which n = (1,2, .... n > is the ground set and S is a family of subsets of n called edges. If every edge in S is an r-element subset of n then H is said to be r-uniform. A graph is a 2-uniform hypergraph. An arrangement or permutation of n is a linear array TV= a, a2 . . . a, of all n points in n. The initial segmentsof TVare 0, (al >, (a,, a2), .... n. A set R of permutations of n realizes H = (n, S) if for every nonempty edge A in S an initial segment s, can be chosen for each Q E R so that A=
f-j
S,.
A moment’s reflection shows that R realizes H if and only if, for every A E S and all x E n\A, x follows an initial segment s of some 0 E R for which A c s. Moreover, if A and B are in the family of subsets obtained as intersections of initial segments of R, then A n B is also in this family. Note that the trivial hypergraph (n, 0) is realized by the empty set of permutations. DEFINITION. The dimensionD(H) of H = (n, S) is the smallest m > 0 for which there exists a,set R of m permutations of n which realizes H. Clearly different at right elements.
D(n, 0) = 0, o(n, (0 >) = 1, and, by any n permutations with last elements, D
