Hypergraph Seminar: Ohio State University 1972

294 Pages·1974·4.219 MB·English-French

by  Claude Berge C.N.R.S. (auth.), Claude Berge, Dijen Ray-Chaudhuri (eds.)

Lecture Notes ni
Mathematics
detidE yb .A Dold dna .B nnamkcE
114
hpargrepyH ranimeS
Ohio State University 1972
detidE yb Berge Claude dna Dijen iruhduahC-yaR
III
Springer-Verlag
Berlin.Heidelberg • New York 4 7 19
Claude Berge
Universit6 de Paris VI
Laboratoire de Calcul des Probabilites
9, Quai St. Bernard - Tour 56
Paris 5e./France
Dijen Ray-Chaudhuri
Ohio State University
Dept. of Mathematics
Colombus Ohio 43210/USA
Library of Congress Cataloging in Publication Data
Working Seminar on H~pergraphs, Ohio State University,
1972.
Hypergraph seminar.
(Lecture notes in mathematics, 411)
Bibliography: p.
.i Hypergraphs--Congresses. .I Berge, Claude, ed.
.iI Ray-Chaudhuri, Dijen, 1933- .de
III. Title. IV. Series.
QAI66.W67 1972 511'.5 74-17434
AMS Subject Classifications (1970): 05-02, 05 B 25, 05 C 15
ISBN 3-540-06846-5 Springer-Verlag Berlin • Heidelberg • New York
ISBN 0-38?-06846-5 Springer-Verlag New York • Heidelberg • Berlin
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© by Springer-Verlag Berlin • Heidelberg 1974. Printed in Germany.
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FOREWORD
This volume constitutes the proceedings of the first Working Seminar
on Hypergraphs, held at the Ohio State University in Columbus (Ohio), from
August 61 to September 9, 1972. The Department of ~athematics invited a
large number of combinatorialists (from Canada, France, Iceland, Hungary
and U.S.A.) to present and discuss results related to the new area of
Hypergraph Theory. Since Hypergraphs can be viewed as generalizations of
Graphs, or of Matroids, or of Finite Geometries, the main task was to unify
concepts and terminology.
We especially thank Professor Arnold E. Ross, Chairman of the Depart-
ment of Mathematics at OSU, for his encouragement and counsel before and
during the conference. His enthusiasm for thisnew field of research and
the constant support he gave to the young mathematicians were a great source
of inspiration for all of us. We would also like to thank U.S.R. Murty,
R.P. Gupta, K.I. Chang, N.R. Grabois and Alan P. Sprague for their help
in editing this volume.
Claude BERGE Dijen K. RAY-CHAUDHURI
This Volume is dedicated
to
PROFESSOR ARNOLD ROSS
TABLE OF CONTENTS
PART I : GENERAL HYPERGRAPHS
C. BERGE
Isomorphism Problems for Hypergraphs . . . . . . . . . . . I
C. BERGE
Nombres de coloration de l'hypergraphe h-parti complet . . 13
C. BERGE and M° SIMONOVITS
The Coloring Numbers of the Direct Product of ~wo
Hypergr aphs . . . . . . . . . . . . . . . . . . . . . . . 21
J.C. BERMOND
Graphe repr~sentatif de l'hypergraphe h-parti complet . .
R° BONNET and P. ERDOS
The Chromatic Index of an Infinite Complete Hypergraph :
a Partition Theorem . . . . . . . . . . . . . . . . . . . 84
LATAVHC
V.
Intersecting Families of Edges in Hypergraphs Having
the Hereditary Property . . . . . . . . . . . . . . . . . 61
W.H. CUNN INGHAM
On theorems of Berge and Fournier . . . . . . . . . . . . 67
SODRE
P.
Extremal Problems on Graphs and Hypergraphs ....... 75
V. FABER
Hypergr aph Reconstruction . . . . . . . . . . . . . . . . 85
J.C. FOURNIER
Une condition pour qu'un hypergraphe, ou son eompl~mentaire,
soit fortement isomorphe ~ un hypergraphe complet .... 95
P. HANSEN and M. LAS VERGNAS
On a Property of Hypergraphs with no cycles of Length
Greater than 2 . . . . . . . . . . . . . . . . . . . . . . 99
M. LAS VERGNAS
Sur les hypergraphes bichromatiques . . . . . . . . . . . 102
s
L. LOVASZ
Minimax Theorems for Hypergraphs . . . . . . . . . . . . . 111
J.C. MEYER
Quelques probl~mes concernant les cliques des hypergraphes
h-complets et q-parti h-complets . . . . . . . . . . . . . 127
R. RADO
Reconstruction Theorems for Infinite Hypergraphs ..... 140
M. SIMONOVITS
Note on a Hypergraph Extremal Problem . . . . . . . . . . 147
F. STERBOUL
i
Sur une conjecture de V. Chvatal . . . . . . . . . . . . 152
F. STERBOUL
On the Chromatic Number of the Direct Product of
Hypergraphs . . . . . . . . . . . . . . . . . . . . . . . 165
PART II.: GRAPHS, MATROIDS, DESIGNS
t
V. CHVATAL and L. LOVASZ
Every Directed Graph Has a Semi-kernel .......... 175
T.A. DOWLING and D°G. KELLY
Elementary Strong Maps and Transversal Geometries .... 176
p. m ~)IR S
Some Problems in Graph Theory . . . . . . . . . . . . . . 187
T, HELGASON
Aspects of the Theory of Hypermatroids .......... 191
W. PULLEYBLANK and Jack EDMONDS
Facets of ]-Matching Polyhedra . . . . . . . . . . . . . . 214
W.T. TUTTE
Chromials . . . . . . . . . . . . . . . . . . . . . . . . 243
R.M. WILSON
Some Partitions of All Triples into Steiner Triple SystemS 267
PART III
UNSOLVED PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . 278
LIST OF SYMBOLS
H = IX,T) = (Ei/i E I) = hypergraph with vertex set X =UE i
h K complete h-uniform hypergraph with n vertices
n
L(H) line graph of H : simple graph representlng the edges of H
R(H) representing multigraph ot H
H dual o~ H
A H subhypergraph of H induced Dy A~ X
(Ei/i ~ J) partial hypergraph of H generated by J c I
H × A section hypergraph of H by A ~ X
n(H) order of H
r(H) rank of H
X(H) weak chromatic number : no edge with more than one element is
monochromatic
y(H) strong chromatic number : no two vertlces in the same edge have the
same color
p~H) weak stability number : maximum cardinality of a set ~ S X containing
no edge with more than one element
~(H) strong stability number : maximum cardinality o~ a set ~ S X such
that ~ S Eil ! I ] for all .i
T[H) transversal number : minimum cardinality of a set ~ T X such that
~ IT EiJ ~ | for all i
~(H) matching number : maximum cardinality ot a family o~ pairwlse disjoint
edges = e(H ~)
q(H) chromatic index = y~H ~)
p(H) covering number = ~(H ~)
6(H) maximum degree = r(H*)
ISOMORPHISM PROBLEMS FOR HYPERGRAPHS
Claude Berge~ C.N.R.S.
]. Introduction. A hypergraph H = (X,~) = ~ : j N~ (Ei : of vertices•= iThe eM)setsis
a family ~ of subsets i E of a set X ={xj (EI'E2''~''Em)
i E are called edges•
The rank r(H) of a hype~graph H is the maximum cardinality of the
edges. If all edges have the same cardinality, the hypergraph is said to be
uniform. The subhypersraph induced by a subset A of X is the hypergraph
A = H (E i~ A : i~M, E ioA # ~)
If I~ M , the partial hypergraph generated by I is the hypergraph
E( i ~ : i )I • The section hypergraph is the partial hypergraph
X H A i = (E : i&M, Ei_~ A ~ X)
The dual ~ H of H is a hypergraphwith vertex set. E = le],...,em~ ,
and having edges which are certain subsets of E ~ namely edges X. where
X. = {ei : iEM, xj~Ei} ]
J
Consider two hypergraphs H = (E 1,...,Em) and H' = F( I,...,F m)
H is equivalent to H' (H = H') if the mapping ~ : X >-- Y , ~(x )i = Yi j
satisfies ~ ~(Ei) = F~i (i~M) for some permutation "~ of M
H is e gual to H' (or H = H') if the permutation Tv in the above
definition can be the identity•
H is isomorphic to H' (or H~ H') if there is a bijection
: X >--- Y and if there is a permutation -77 of M such that
~(E )i = F i (i~M) The bijection ~ is called an isomorphism.
H is strongly isomorphic to H' (or H ~ H') if there is a bijection
X : >-- Y for which ~ (E )i i = F for all i~M
Observe that equality implies the other three relations and any of the
relations implies isomorphism.
We give several examples:
:elpmaxE. Consider the following:
H =
Observe that H _~ 'H but , H~ ! 'H , since if H_.~ 'H , the
vertex x would map to the nonexistent vertex meeting edges 1,2, and
in 'H
Example: Consider the line graph L(H) of the graph H above:
L(H) = ~ ~ ~
9
Observe that L(H) = L(H') , but since the edges are unlabeled here,
equality is meaningless.
Our purpose in this paper is to present some general results concerning
isomorphisms and other relations among hypergraphs.
A multigraph is a hypergraph with IEil ~ 2 for all i~M
Proposition °I If H = ,X( (Ei)i~M) and 'H = ,Y( (Fi)i~ ) M are
multigraphs, and if ~ X : ~-- Y is a bijection, then the following are
equivalent:
)i( ~ is an isqmO~hism;
(ii) mH(x,y) = mH,(~(x), ~(y)) for all x,y e X , where
mH(x,y) is the number of edges ~dining x d~__na 'x ni--
Proposition .2 If H is a hypergraph, then (H ~)q~ = H
Proposition .3 If H, 'H are hypergraphs, then ~ H 'H if and only
if ~ ~ H H '~
Proposition .4 If ,H 'H are hypergraphs, then ~ H 'H if and only if
_~=~H ,~ H
Proposition .5 The dual of the partial hypergraph of H generated by
E( i : ie )I equals the subhypergraph of ~ H induced by {e i : i~ 13
Proposition ~ 5 The dual of the subhypergraph
HA = ~tiE( A : iEM, Ei~A # )~ with A = {xj : j~ J} equals the partial
h ypergraph of H ~A generated by jX( E j : )J
Proposition .6 The dual of the section hypergraph X H A, A I = jx J~, 6 j :
equals the subhypergraph of H "~ induced by U Xj- ~ .X
N ~ j j£N-J 3