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On the numbers of independent k-sets in a claw free graph
JOURNAL
OF COMBINATORIAL
THEORY,
B 50, 241-244 (1990)
SERIES
On the Numbers of independent in a Claw Free Graph YAHYA
UER
48-ER
Universitk Combinatoire,
OULD HAMIDOUNE Pierre et Marie Curie, 4 Place Jussieu, 75230 Paris,
Communicated Received
Let V(C) hence result
k-Sets
France
by the Editor
February
9, 1988
G be a graph without induced K,,,. The number of independent k-subsets of will be denoted by sk, We prove that the sequence (So) is log concave and unimodal. These results are known for line graphs as a consequence of a It' 1990 Academic Press. Inc. of Heilman and Lieb.
1. INTRODUCTION Let G be a graph. The vertex set (resp. edge set) of G will be denoted by V(G) (resp. E(G)) or simply V (resp. E). The set of vertices adjacent to a vertex x will be denoted by T(x). When A c V(G), the subgraph of G induced by A will be denoted by G[A 1. We sometimes write A instead of G[A]. A set with cardinality k will be called a k-set. In particular a matching of size k will be called a k-matching. A subset S of V(G) is called independent if for every x E S, T(x) n S = 121. The maximal cardinality of an independent set is called the independence number of G and will denoted by a(G). The set (resp. number) of independent k-subsets of V(G) will be denoted by Sk(G) (resp. sk (G)) or simply Sk (resp. sk). The number of k-matchings of G will be denoted by mk (G). We recall that G is called claw free if it contains no induced K,.,, i.e., for every vertex XE V(G), cr(G[T(x)]) < 2. Let G be a graph. The line graph of G, denoted by L(G), is the graph with vertex set E, where distict edges of G are adjacent if they have a common vertex. It is obvious that L(G) is a claw free graph. Some results of matching are extended to independent sets in claw free graphs (cf. [3]). Let r be the maximum cardinality of a matching. It was proved by Heilman and Lieb [2] that rn: 2 (1 + l/k)( 1 + l/(r - k)) mk + 1mk ~, , In fact, they proved the following stronger result. 241 0095-8956/90
$3.00
Copyright (Q 1990 by Academic Press, Inc. All rights of reproductmn I" any form reserved.
242
YAHYAOULDHAMIDOUNE
THEOREM (Heilman and C( - 1)kmk X2k are all real.
Lieb
[2]).
The roots
of
the polynomial
Another proof is contained in Godsil [ 11. This sequence is also considered in the survey paper of Stanley [6]. The sequence (sk) is not necessarily unimodal as mentioned by Mason c41. We prove that for a claw free graph, sz 2 (1 + I/k) Sk+ isk _ , + Sk. Let H be the line graph of G. Then clearly m,(G) = s,(H). Our bound in this case is weaker than the above bound of Heilman and Lieb. Our method gives a combinatorial proof for the log concavity of the sequence (mk). The unimodality of this sequence is proved combinatorially by Schwenk [S]. We know of no claw free graph such that C skxk has a nonreal root. If all the roots of the above polynomial are real then by Newton’s inequalities, s: > (1 + l/k)( 1 + l/(a - k)) Sk+, Sk_, , where u = a(G). We conjecture this last inequality for claw free graphs. This relation is true for line graphs by the above result of Heilman and Lieb. 2. CLAW FREE GRAPHS A subset S of V(G) is said to be doubly independent if S and I’(G) - S are both independent. The set of doubly independent k-subsets of V(G) will be denoted by Yk(G). We write tik(G)= lul,(G)l. A subgraph G’ of G is said to be even (resp. odd) if 1V(G’)I is even (resp. odd). Let G be a graph which is either an even cycle or a path. We see easily that G contains exactly two doubly independent sets. Any one of these sets will be called an extremity. When G is odd the biggest extremity will be called initial. LEMMA 2.1. Let c (resp. 2m) be the be an integer such ticular for any claw
G be a claw free bipartite graph with even order 2n. Let number of even (resp. odd) components of G, and let j that 0 < j < 2m. Then $,, + mu ,(G) = 2’( ,,$‘,). In parfree graph with order 2k, $k (G) 3 (1 + l/k) tik _ , (G).
Proof. Suppose G is bipartite and let x be a 2 vertex of G. Since cr(r(x)) < 2, we have d(x) < 2. It follows that all the connected components of G are paths or even cycles. In order to form a doubly independent (n + m -j)-set with respect to G, we need just to choose one extremity in each even component and 2m - j initial extremities among the odd components. It follows that
*n+m-j(G)=2’
243
ON THE NUMBERS OF INDEPENDENT SETS
The last part of the lemma follows from (1) and from the obvious equality It/k- i (G) = 0 when G is nonbipartitte. 1 Remark. The (1 +~/cz)J/~-~(G),
above proof shows where (r=a(G).
the
better
bound
I,$~(G) b
Our proof compares s: to sk+, sk-, by partitioning S, x Sk and s k+lxSk-I. We admit partitions with void terms. We describe this partition as follows. Let A c V and B c A. We put
P~,~= {(A’, Y) 1 Xv Y=A and Xn Y= B). Let A c V such that
IAl = 2k. It is clear from the definitions
I~~n(S,xS,)l=~,(GCAl)and
that
IPAn(S,-,xS,+,)I=~,-,(GCAI).
2.2. Let A c V such that )Al = 2k - j and B c A such that IBI =.i U’PA,B~(S,-,XS,+, ) # @ then A is the union of an independent (k + l)-set and an independent (k - l)-set both containing B. LEMMA
Proof
Immediate
from the definitions.
LEMMA
2.3. Let A c V such that 1Al = 2k -j
lBl=j.Zf~~.~n(S,-,xS,+,)Z~~ b/,.Bn(S,xS,)I
1 and Bc A such that
then
=$k-,(GCA-Bl)
and
lPA,B n(Sk-,xSk+Ijl=~k-,-,(GCA-Bl). Proof.
The proof is easy. 1
LEMMA
2.4. Let G be a claw free graph. Then
C IPAn((Skx&)l. IAl =i
Let j=2k-i and let IAl =i. Then {pA,Bn(Sk--I~Sk+l):BCA and lBl=j} is a partition of pa n (S, _, x S, + , ). Similarly for pa n (S, x S,). By Lemma 2.1, Lemma 2.3, and the trivial inequality 1 + l/k < 1 + l/(k - j), we have ProoJ
(1 + l/k)
1PA.B
“(S~-IXSk+l)~~~PA,Bn(S~xS~)~~
244
YAHYA
OULD
HAMIDOUNE
By taking the sum of both sides, B c A and IBI = j and IAl = i, we get the desired relation. 1 2.5.
PROPOSITION
Let G be a claw free graph. Then s~+~(G)s~-~(G)+s~(G).
Proof
G
E
1
;=I+& IAl=r =JS,xS,-AI=s:-sk
using the above lemma,
bAn(SkxSk)I
where
A= {(X,X)
11~s~).
1
Conjecture. s~~(1+~)(1+~)s,_,s,,,
where
cr=a(G).
REFERENCES 1. C. D. GODSIL, Real graph polynomials, in “Progress in Graph Theory” (5. A. Bondy and U. S. R. Murty, Eds.), pp. 281-293, Academic Press, Toronto/Orlando, 1984. 2. 0. J. HEILMAN AND E. H. LIEB, Theory of monomer-dimer systems, Comm. Math. Phys. 25 (1972), 19&232. 3. L. LOVASZ AND P. PLUMMER, Matching theory, Akadtmiai Kiado, Budapest, 1986; also published as North-Holland Mathematical Studies, Vol. 121, North-Holland, Amsterdam, 1986. 4. J. H. MASON, Unimodal conjectures and Motzkin’s theorem, in “Combinatorics” (Proc. Conf. Combinatorial Math., Math. Inst. Oxford, 1972), pp. 207-220, Inst. Math. Appl., Southen on Sea, 1972. 5. A. J. SCHWENK, On unimodal sequences of graphical invariants, J. Combin. Theory Ser. B 30 (1981), 247-250. 6. R. P. STANLEY, Log-concave and unimodal sequances in algebra, combinatorics and geometry, preprint.
OF COMBINATORIAL
THEORY,
B 50, 241-244 (1990)
SERIES
On the Numbers of independent in a Claw Free Graph YAHYA
UER
48-ER
Universitk Combinatoire,
OULD HAMIDOUNE Pierre et Marie Curie, 4 Place Jussieu, 75230 Paris,
Communicated Received
Let V(C) hence result
k-Sets
France
by the Editor
February
9, 1988
G be a graph without induced K,,,. The number of independent k-subsets of will be denoted by sk, We prove that the sequence (So) is log concave and unimodal. These results are known for line graphs as a consequence of a It' 1990 Academic Press. Inc. of Heilman and Lieb.
1. INTRODUCTION Let G be a graph. The vertex set (resp. edge set) of G will be denoted by V(G) (resp. E(G)) or simply V (resp. E). The set of vertices adjacent to a vertex x will be denoted by T(x). When A c V(G), the subgraph of G induced by A will be denoted by G[A 1. We sometimes write A instead of G[A]. A set with cardinality k will be called a k-set. In particular a matching of size k will be called a k-matching. A subset S of V(G) is called independent if for every x E S, T(x) n S = 121. The maximal cardinality of an independent set is called the independence number of G and will denoted by a(G). The set (resp. number) of independent k-subsets of V(G) will be denoted by Sk(G) (resp. sk (G)) or simply Sk (resp. sk). The number of k-matchings of G will be denoted by mk (G). We recall that G is called claw free if it contains no induced K,.,, i.e., for every vertex XE V(G), cr(G[T(x)]) < 2. Let G be a graph. The line graph of G, denoted by L(G), is the graph with vertex set E, where distict edges of G are adjacent if they have a common vertex. It is obvious that L(G) is a claw free graph. Some results of matching are extended to independent sets in claw free graphs (cf. [3]). Let r be the maximum cardinality of a matching. It was proved by Heilman and Lieb [2] that rn: 2 (1 + l/k)( 1 + l/(r - k)) mk + 1mk ~, , In fact, they proved the following stronger result. 241 0095-8956/90
$3.00
Copyright (Q 1990 by Academic Press, Inc. All rights of reproductmn I" any form reserved.
242
YAHYAOULDHAMIDOUNE
THEOREM (Heilman and C( - 1)kmk X2k are all real.
Lieb
[2]).
The roots
of
the polynomial
Another proof is contained in Godsil [ 11. This sequence is also considered in the survey paper of Stanley [6]. The sequence (sk) is not necessarily unimodal as mentioned by Mason c41. We prove that for a claw free graph, sz 2 (1 + I/k) Sk+ isk _ , + Sk. Let H be the line graph of G. Then clearly m,(G) = s,(H). Our bound in this case is weaker than the above bound of Heilman and Lieb. Our method gives a combinatorial proof for the log concavity of the sequence (mk). The unimodality of this sequence is proved combinatorially by Schwenk [S]. We know of no claw free graph such that C skxk has a nonreal root. If all the roots of the above polynomial are real then by Newton’s inequalities, s: > (1 + l/k)( 1 + l/(a - k)) Sk+, Sk_, , where u = a(G). We conjecture this last inequality for claw free graphs. This relation is true for line graphs by the above result of Heilman and Lieb. 2. CLAW FREE GRAPHS A subset S of V(G) is said to be doubly independent if S and I’(G) - S are both independent. The set of doubly independent k-subsets of V(G) will be denoted by Yk(G). We write tik(G)= lul,(G)l. A subgraph G’ of G is said to be even (resp. odd) if 1V(G’)I is even (resp. odd). Let G be a graph which is either an even cycle or a path. We see easily that G contains exactly two doubly independent sets. Any one of these sets will be called an extremity. When G is odd the biggest extremity will be called initial. LEMMA 2.1. Let c (resp. 2m) be the be an integer such ticular for any claw
G be a claw free bipartite graph with even order 2n. Let number of even (resp. odd) components of G, and let j that 0 < j < 2m. Then $,, + mu ,(G) = 2’( ,,$‘,). In parfree graph with order 2k, $k (G) 3 (1 + l/k) tik _ , (G).
Proof. Suppose G is bipartite and let x be a 2 vertex of G. Since cr(r(x)) < 2, we have d(x) < 2. It follows that all the connected components of G are paths or even cycles. In order to form a doubly independent (n + m -j)-set with respect to G, we need just to choose one extremity in each even component and 2m - j initial extremities among the odd components. It follows that
*n+m-j(G)=2’
243
ON THE NUMBERS OF INDEPENDENT SETS
The last part of the lemma follows from (1) and from the obvious equality It/k- i (G) = 0 when G is nonbipartitte. 1 Remark. The (1 +~/cz)J/~-~(G),
above proof shows where (r=a(G).
the
better
bound
I,$~(G) b
Our proof compares s: to sk+, sk-, by partitioning S, x Sk and s k+lxSk-I. We admit partitions with void terms. We describe this partition as follows. Let A c V and B c A. We put
P~,~= {(A’, Y) 1 Xv Y=A and Xn Y= B). Let A c V such that
IAl = 2k. It is clear from the definitions
I~~n(S,xS,)l=~,(GCAl)and
that
IPAn(S,-,xS,+,)I=~,-,(GCAI).
2.2. Let A c V such that )Al = 2k - j and B c A such that IBI =.i U’PA,B~(S,-,XS,+, ) # @ then A is the union of an independent (k + l)-set and an independent (k - l)-set both containing B. LEMMA
Proof
Immediate
from the definitions.
LEMMA
2.3. Let A c V such that 1Al = 2k -j
lBl=j.Zf~~.~n(S,-,xS,+,)Z~~ b/,.Bn(S,xS,)I
1 and Bc A such that
then
=$k-,(GCA-Bl)
and
lPA,B n(Sk-,xSk+Ijl=~k-,-,(GCA-Bl). Proof.
The proof is easy. 1
LEMMA
2.4. Let G be a claw free graph. Then
C IPAn((Skx&)l. IAl =i
Let j=2k-i and let IAl =i. Then {pA,Bn(Sk--I~Sk+l):BCA and lBl=j} is a partition of pa n (S, _, x S, + , ). Similarly for pa n (S, x S,). By Lemma 2.1, Lemma 2.3, and the trivial inequality 1 + l/k < 1 + l/(k - j), we have ProoJ
(1 + l/k)
1PA.B
“(S~-IXSk+l)~~~PA,Bn(S~xS~)~~
244
YAHYA
OULD
HAMIDOUNE
By taking the sum of both sides, B c A and IBI = j and IAl = i, we get the desired relation. 1 2.5.
PROPOSITION
Let G be a claw free graph. Then s~+~(G)s~-~(G)+s~(G).
Proof
G
E
1
;=I+& IAl=r =JS,xS,-AI=s:-sk
using the above lemma,
bAn(SkxSk)I
where
A= {(X,X)
11~s~).
1
Conjecture. s~~(1+~)(1+~)s,_,s,,,
where
cr=a(G).
REFERENCES 1. C. D. GODSIL, Real graph polynomials, in “Progress in Graph Theory” (5. A. Bondy and U. S. R. Murty, Eds.), pp. 281-293, Academic Press, Toronto/Orlando, 1984. 2. 0. J. HEILMAN AND E. H. LIEB, Theory of monomer-dimer systems, Comm. Math. Phys. 25 (1972), 19&232. 3. L. LOVASZ AND P. PLUMMER, Matching theory, Akadtmiai Kiado, Budapest, 1986; also published as North-Holland Mathematical Studies, Vol. 121, North-Holland, Amsterdam, 1986. 4. J. H. MASON, Unimodal conjectures and Motzkin’s theorem, in “Combinatorics” (Proc. Conf. Combinatorial Math., Math. Inst. Oxford, 1972), pp. 207-220, Inst. Math. Appl., Southen on Sea, 1972. 5. A. J. SCHWENK, On unimodal sequences of graphical invariants, J. Combin. Theory Ser. B 30 (1981), 247-250. 6. R. P. STANLEY, Log-concave and unimodal sequances in algebra, combinatorics and geometry, preprint.