Table Of ContentLecture Notes ni
Mathematics
Edited yb .A Dold dna .B Eckmann
885
scirotanibmoC
dna Graph yroehT
Proceedings of the Symposium Held ta
the Indian Statistical Institute, Calcutta,
February 25-29, 1980
Edited yb .S .B oaR
galreV-regnirpS
nilreB Heidelberg New kroY 1891
Editor
Siddani Bhaskara Rao
Indian Statistical Institute
203, Barrackpore Trunk Road
Calcutta - 700035, India
AMS Subject Classifications (1980): 05-02, 05-04, 05-06, 05 A15,
05 Bxx, 05 Cxx, 62K10, 68E10
ISBN 3-540-11151-4 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-11151-4 Springer-Verlag NewYork Heidelberg Berlin
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ECAFERP
This volume consists of the Proceedings of the second Symposium on Combinatorics
and Graph Theory held at the Indian Statistical Institute, s from February 25 to
February 29, 1980 under the auspices and with the full financial support of the Indian
Statistical Institute. I would like to express ym gratitude to Professor B.P. Adhikari,
Director of the Institute, Professor A. Maitra, the then Chairman, Division of Theoreti-
cal Statistics and Mathematics and the faculty of the Division for giving em the
opportunity to convene the symposium and for their valuable advice and suggestions
throughout.
The aim of the symposium was to discuss the recent trends in Combinatorics and
Graph Theory and to expose the young eombinatorialists and graph theorists in India
to the recent developments in these areas.
eW would like to thank the special invited speakers - Professors C. Berge (France),
P. Erd~s (Hungary), E.V. Krishnamurthy (India), L. Lov~sz (Hungary), K.R. Parthasarathy
(India), D.K. Ray-Chaudhuri (USA), 3.3. Seidel (The Netherlands), S.S. Shrikhande (India),
Vera T. S6s (Hungary) and M.N. Vartak (India) - for their thought-provoking special
invited addresses, active participation in the deliberations of the symposium and for
their valuable contributions to these proceedings (Professor L. Lov~sz was unable to
attend the symposium for personal reasons, but we are glad to include his contribution
as a part of the proceedings). eW thank all the participants who have chaired the
sessions.
All the contributed papers appearing in this volume were refereed. eW thank all
the referees for their help in this regard. eW also thank all the participants for
their keen interest in the symposium and for their interesting contributions to these
proceedings. Both the special invited addresses and the contributed papers are arranged
alphabetically according to the authors' names.
I would like to use this opportunity to thank most warmly all those, apart from
the special invited speakers and the participants, who have contributed very much to
the success of the symposium: colleagues at the Indian Statistical Institute, parti-
cularly those of the Stat-Math Division, Dean's office, Director's office, Eka press,
Electrical and Engineering units, Estate office, Guest House, Library, Hedieal Welfare
unit, Reprography, Photography and Audio Visual units, Transport and Travel Cell, and
more particularly Dr.(Hrs.) Hirmala Achuthan, Dr. A. Ramachandra Rao, Dr. T.3. Rao,
Hr. N. Srinivasan, Dr. K.S. Vijayan, Ph.D. students Mr. Prabir Das, Hr. T. Gangopadhyay
and graduate students Mr. Atul 3ain and Mr. Rahul Roy for their active participation,
valuable help and suggestions in the various organizational committees.
VI
I would like to express ym gratitude to Professor J.K. Ohosh, present namriahC
of the Stat-Math Division~for his advice dna for providing all the facilities at his
disposal. I would like to express ym gratitude to Dr. .A ardnahcamaR oaR under esohw
excellent supervision the typing of the manuscript saw done, including the difficult
task of proof reading the entire manuscript, with the cooperation of Dr.(Mrs.) Nirmala
Achuthan, .rM Prabir saD dna .rM .T yayhdapognaO while I saw away visiting the depart-
tnem of mathematics, ehT Ohio State University, submuloC )ASU( during the cimedaca
year 1980-81. I ma immensely grateful to them for their kind help in this regard.
I thank .rM Arun saD for the elegant typing of these proceedings. Thanks are
also eud to .rM S.K. eD for the excellent diagrams.
Last not least I thank .rM .K Kundu, .rM .S atpuGsaD dna .rM .P Nandi for
secretarial help.
February ,91 1891 INADDIS ARAKSAHB OAR
STNETNOC
Special Invited Addresses
Diperfeot Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.C Berge
emoS wen problems dna results in Graph Theory
dna other branches of Combinatorial Mathematics . . . . . . . . . . . . . . . .
.P ErdBs
A form invariant multivariable polynomial representation of graphs . . . . . . . 81
E.V. Krishnamurthy
emoS combinatorial applications of the wen linear programming algorithm .... 33
L. Lov~sz dna .A Schrijver
In search of a complete invaKiant for graphs . . . . . . . . . . . . . . . . . . 24
.K Balasubramanian dna K.R. Parthasarathy
Affine triple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 06
D.K. Ray-Chaudhuri
Tables of two-graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 07
F.C. Bussemaker, R.A. Mathon dna 3.J. Seldel
Designs, adjacency multigraphs dna sgniddebme : a survey . . . . . . . . . . . . 311
S.S. Shrikhande dna N.M. Singhi
nO the adjugate of a symmetrical balanced incomplete block design with I = 1 331
G.A. Patwardhan dna M.N. Vartak
Contributed Papers
Characterization of potentially connected integer-pair sequences . . . . . . . . 351
Nirmala Achuthan
Construction dna combinatorial properties of othogonal arrays
with variable rebmun of symbols in rows . . . . . . . . . . . . . . . . . . . . 561
.B Adhikary dna Premadhis saD
Construction of group divisible rotatable designs . . . . . . . . . . . . . . . 171
.B Adhikary dna .R adnaP
emoS path-length properties of graphs dna digraphs . . . . . . . . . . . . . . . 581
.G.W Bridges
2-2 perfect graphi c degree sequences . . . . . . . . . . . . . . . . . . . . . . 591
P.D. ehtawahC
Characterization of forcibly outerplanar graphic sequences . . . . . . . . . . . 302
S.A. muduohC
IV
Characterization of potentially self-complementary,
self-converse degree-pair sequences for digraphs . . . . . . . . . . . . . . . . 212
Prabir saD
Set-reconstruction of chain sizes in a class of finite topologies . . . . . . . 722
nwapwahS .K saD
Characterization of forcibly bipartite self-complementary
bipartitioned sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732
.T yayhdapognaG
A graph theoretical recurrence formula for computing
the characteristic polynomial of a matrix . . . . . . . . . . . . . . . . . . . 162
.K.M Gill
A note concerning Acharya's conjecture no a spectral measure
of structural balance in a social system . . . . . . . . . . . . . . . . . . . . 662
.K.M Gill
nO permutation-generating strings and rosaries . . . . . . . . . . . . . . . . . 272
Hansraj Gupta
Enumeration of labelled digraphs and hypergraphs . . . . . . . . . . . . . . . . 276
Malati Hegde and M.R. Sridharan
Analysis of a spanning tree enumeration algorithm . . . . . . . . . . . . . . . 284
.R 3ayakumar and K. Thulasiraman
Binding number, cycles and complete graphs . . . . . . . . . . . . . . . . . . . 092
[email protected] Kane and S.P. Mohanty
A class of counterexamples to a conjecture no diameter critical graphs ..... 792
.V Krishnamoorthy and .R ramukadnaN
Reconstruction of a pair of connected graphs from their line-concatenations . . 103
V.R. Kulli and N.S. Annigeri
nO domination related concepts in Graph Theory . . . . . . . . . . . . . . . . . 803
uneR Laskar and H.B. Walikar
The local central limit theorem for Stirling numbers of the
second kind dna an estimate for Bell numbers . . . . . . . . . . . . . . . . . . 123
.V.V noneM
A family of hypo-hamiltonian generalized prisms ................ 331
S.P. Mohanty and Daljit Rao
Graphical cyclic permutation groups . . . . . . . . . . . . . . . . . . . . . . 933
S.P. Mohanty, M.R. Sridharan dna S.K. Shukla
Orthogonal main effect plans with variable number of levels for factors .... 347
A.C. Mukhopadhyay
nO molecular and atomic matroids . . . . . . . . . . . . . . . . . . . . . . . . 358
.H Narayanan and M.N. Vartak
New inequalities for the parameters of an association scheme .......... 365
.A Neumaier
Vll
nO the (4,3)-regular subgraph conjecture . . . . . . . . . . . . . . . . . . . . 863
K.R. Parthasarathy dna .S Srldharan
Enumeration of Latin rectangles via SDR's . . . . . . . . . . . . . . . . . . . 083
C.R. Pranesachar
Nearly line regular graphs and their reconstruction . . . . . . . . . . . . . . 391
S. Ramachandran
nO reconstructing separable digraphs . . . . . . . . . . . . . . . . . . . . . . 604
.S nardnaheamaR
nO degree sequences of cacti . . . . . . . . . . . . . . . . . . . . . . . . . . 014
.A Ramachandra oaR
A survey of the theory of potentially P-graphic
dna forcibly P-graphic degree sequences . . . . . . . . . . . . . . . . . . . . 417
S.B. oaR
sdrawoT a theory of forcibly hereditary P-graphic sequences . . . . . . . . . . 144
S.B. Rao
The minimal forbidden subgraphs for generalized line-graphs . . . . . . . . . . 459
S.B. Rao, N.M. Singhi and K.S. Vijayan
Spectral characterization of the line graph of &K n . . . . . . . . . . . . . . . 473
S.B. Rao, N.M. SJnghi and K.S. Vijayan
Balanced arrays from association schemes dna some related results . . . . . . . 184
G.M. Saha
Some further combinatorial and constructional aspects . . . . . . . . . . . . . 488
of generalized Youden designs
Bikas K. Sinha
mumixaM degree gnoma vertices of a non-Hamiltonian
homogeneously traceable graph . . . . . . . . . . . . . . . . . . . . . . . . . 694
Z. Skupie#
TCEFREPID SHPARG
CLAUDE BERGE
University of Paris, Paris IV
.1 INTRODUCTION
Gallai and Milgram have shown that in a directed graph, the minimum number of
vertex-disjoint (elementary, directed) paths needed to cover the vertex-set is smaller
than or equal to the stability number (i.e., maximum number of independent vertices).
However, the various proofs of this result do not imply the existence of an optimal
stable set S and of a partition of the vertex-set into paths pl,P2,...,pk such
that pI iA IS = I for all .i
Similarly, Gallai and Roy have shown independently that in a directed graph, the
maximum number of vertices in an (elementary, directed) path is greater than or equal
to the chromatic number; here again, we do not know if there exists on optimal coloring
(SI,S2,...,S )k and a path p such that PI NSil = 1 for all .i
In this paper, we give proofs and applications for the two above mentioned
results; then we study the graphs ~-diperfect and y-diperfect, which suggest several
conjectures similar to the perfect graph conjecture. I wish to thank Polly Underground
who has suggested the proof of Proposition .1
2. EHT MARGLIM-IALLAG MEROEHT DNA EMOS SNOITACILPPA
Let G = (X,U) be a directed graph with vertex set X, and a set U~X x X of
arcs (or "directed edges"). A path will be defined as a sequence of vertices
Xl,X2,...,Xk, all distinct, such that (Xl,X 2) s U, (x2,x 3) E U,...,(Xk_l,Xk) c ;U every
path p = (Xl,X2,...,x k) defines a set that ew denote p = {Xl,X2,...,Xk}.
A family M = {PI' 2P ''''} is a partition of X if the Pi are pairwise
vertex-disjoint paths, and if U Pi = X.
We shall give a proof of the following
Theorem (Gallai-Milgram). In a directed graph G, we have miniMl 2 ~(G). The
M
minimum is taken over all the partitions M of X into paths, and ~(G) denotes the
stability number of G (i.e., the maximum number of independent vertices).
Proof. For a family M ={pl,p2,...}, the initial vertices of the paths pl,P2,...
will be denoted by : A(M) = {al,a2,... .} We prove here a stronger statement, namely :
Given a partition M into paths~ there exists a partition 'M into paths such that
I'MI 2 ~(G) and A(M')CA(M).
This proposition is valid for graphs with 1,2 vertices; assume that it is valid
for all graphs with less than n vertices, and we show that it is also valid for a graph
G with n vertices.
.1 We may assume that lMi = ~(G) + .1 Because, if IM 2 ~(G), the proposi-
tion is proved. If iMI ~ ~(G) + ,l consider the first path Pl in ,M and consider
the subgraph G I of G induced by X - Pl" Since m(G 1) ~ m(G), there exists a parti-
tion M I of G I with A(M1)~A(M ) and IMII 2 m(G 1) ~ ~(G). Hence M' = MIU{Pl}
is a partition of G with A(M')CA(M), iM'I ~ m(G) + 1, and we prove the result for
'M instead of .M
.2 Since IA(M) I = IMI = ~(G) + 1, the set A(M) = {al,a 2 .... } is not stable,
so two vertices of A(M) are linked by an arc, say (al,a )2 r .U
The path Pl = (al'bl'Cl'"') has more than one vertex : otherwise, the proposi-
tion is proved with the partition 'M obtained from M by removing pl,p 2 and adding
(al,a2,b2,c2,...). By the induction hypothesis, the subgraph G 1 induced by X - {a l}
has a partition M 1 with IMII ~ ~(G), A(Ml) C{bl,a2,a3,... }. If b I c A(M1), the
proposition is proved with the partition M' obtained from M 1 by adding a I to the
path of M 1 starting at b 1. If b I ~ A(M 1) and a 2 s A(M1) , add a I to the path
of M I starting at a 2. If b I ~ A(M I) and a 2 r A(M1) , take M' = M 1 U {(al)}. This
proves the theorem.
Note that this result shows that in every graph G there exists a partition
with IMI = ~(G), but it does not show the existence of a stable set S of maximum
size and of a partition M = {pl,P2,...} such that : IS N pil = 1 for all i.
Remark. A stronger resuit has been obtained by Las Vergnas 7: Every quasi-strongly
connected graph G contains a spanning arborescenee with at most ~(G) terminal vertices
("sinks"). For a proof, see 9.
The Gallai-Milgram Theorem for a graph G can be obtained by using the Las Vergnas
Theorem for a graph obtained from G by adding a new vertex Xo, that we join with every
vertex of G by an arc going out of x .
o
The appffications of the Theorem of Gallai-Milgram are interesting and numerous.
Application l.(Theorem of REdei). If G = (X,U) is a tournament (complete antisymmetric
graph)~ there exists a path which meets each vertex exactly once.
This follows from the theorem, since we have for G :
minim I ~ ~(G) = .1
Application 2.(Theorem of Dilworth). If G is a transitive graph (i.e., (x,y) ~ U,
(y,z) s U implies (x,z) ~ U), then minim I = ~(G).
If 9(G) denotes the least number of cliques needed to cover X, we have ~(G) ~ 9(G) =
minlMi ~ ~(G).
Application 3.(Theorem of Erd~s-Szekeres). Let n,p,q be positive integers with
n > pq, and let a = (al,a2,...,a n) be a sequence of n distinct integers. Then there
exists either a decreasing subsequence of o with more than p integers~ or an increas-
ing subsequence of a with more than q integers.
Let O = (X,U) be the graph with X = {al,a2,...~an}~ where (ai,a j) c U iff i < j
and a i < aj. If {pl,P2,...,} is a partition of X into ~(G) paths,
lpJxam.)G(~ ~ zlpil = n > pq.
So, we have either ~(G) > p and there exists a decreasing subsequenee with
~(G) > p integers - or maxlpl > q and there exists an increasing subsequence
with more than q integers.
3. THE GALLAI-ROY THEOREM AND SOME APPLICATIONS
Now, we shall consider similar problems for the chromatic number y(G) of a
directed graph G. First, we shall give a simple proof for the following
