Bounds for graph invariants
arXiv:math/0510387v2 [math.CO] 29 Nov 2005
Isidoro Gitler
1
Departamento de Matem´ aticas Centro de Investigaci´on y de Estudios Avanzados del IPN Apartado Postal 14–740 07000 M´exico City, D.F. e-mail
[email protected]
Carlos E. Valencia
2
Instituto de Matem´ aticas Universidad Nacional Autonoma de M´exico Circuito Exterior, Ciudad Universitaria M´exico City, D.F., 04510, M´exico e-mail
[email protected]
Abstract Let G be a graph without isolated vertices and let α(G) be its stability number and τ (G) its covering number. The σv -cover number of a graph, denoted by σv (G), is the maximum natural number m such that every vertex of G belongs to a maximal independent set with at least m vertices. In the first part of this paper we prove that α(G) ≤ τ (G)[1 + α(G) − σv (G)]. We also discuss some conjectures analogous to this theorem. In the second part we give a lower bound for the number of edges of a graph G as a function of the stability number α(G), the covering number τ (G) and the number of connected components c(G) of G. Namely, let a and t be two natural numbers and let ( a ) X zi Γ(a, t) = min | z1 + · · · + za = a + t and zi ≥ 0 ∀ i = 1, . . . , a . 2 i=1
Then if G is any graph, we have: |E(G)| ≥ α(G) − c(G) + Γ(α(G), τ (G)).
1
Preliminaries
Let G = (V (G), E(G)) be a graph with |V | = n vertices and |E| = q edges. Given a subset U ⊂ V (G), the neighbour set of U , denoted by N (U ), is defined as N (U ) = {v ∈ V (G) | v is adjacent to some vertex in U }. A subgraph H is called an induced subgraph, denoted by H = G[V (H)], if H contains all the edges {vi , vj } ∈ E(G) with vi , vj ∈ V (H). A subset M of V is called a stable set if no two vertices in M are adjacent. We call M a maximal stable set if it is maximal with respect to inclusion. The stability number of a graph G is given by
1 2
α(G) = max{|M | | M ⊂ V (G) is a stable set in G}.
The authors where partially supported by CONACyT grants 49091, 49835 and SNI. The author was partially supported by CTIC, UNAM.
1
A subset W of V is called a clique if any two vertices in W are adjacent. We call W maximal if it is maximal with respect to inclusion. The clique number of a graph G is given by ω(G) = max{|W | | W ⊂ V (G) is a clique in G}. The complement of a graph G, denoted by G, is the graph with the same vertex set as G, and edges all pairs of distinct vertices that are nonadjacent in G. Clearly we have that W is a clique of G if and only if W is a stable set of G, therefore ω(G) = α(G). A subset A ⊂ V is a minimal vertex cover for G if: (i) every edge of G is incident with one vertex in A, and (ii) there is no proper subset of A with the first property. If A satisfies condition (i) only, then A is called a vertex cover of G. It is convenient to regard the empty set as a minimal vertex cover for a graph with all its vertices isolated. Note that a set of vertices in G is a maximal stable set if and only if its complement is a minimal vertex cover for G. The vertex covering number of G, denoted by τ (G), is the number of vertices in a minimum vertex cover in G, that is, the size of any smallest vertex cover in G. Thus we have: α(G) + τ (G) = n. The σv -cover number of a graph, denoted by σv (G), is the maximum natural number m such that every vertex of G belongs to a maximal independent set with at least m vertices. We define ωv (G) as σv (G). Recall that a set of edges in a graph G is called independent or a matching if no two of them have a vertex in common. A pairing by an independent set of edges of all the vertices of a graph G is called a perfect matching.
2
B-graphs
The following results are a refinement of the main theorem in [3], in that paper the authors where motivated in bounding invariants for edge rings. In this paper we concentrate only in the combinatorial aspects of these bounds. A graph is called a B-graph if every vertex belongs to a maximum stable set (that is to a stable set of largest size). This concept was introduced by Berge [1]. A graph is τ -critical if τ (G \ v) < τ (G) for all the vertices v ∈ V (G). The following theorem is a bound for invariants of the same type as the ones obtained in [3]. We thank N. Alon (private communication) for some useful suggestions in making the proof of this result simpler and more readable. Theorem 2.1 Let G be a graph without isolated vertices, then α(G) ≤ τ (G)[1 + α(G) − σv (G)]. Proof. First, fix a minimal vertex cover C with τ (G) vertices. Let v ∈ C, then there exist a maximal stable set M ′ with v ∈ M ′ and |M ′ | ≥ σv (G). Hence there exist a 2
natural number k ≤ τ (G) and T1 , . . . , Tk maximal stable sets with |Ti | ≥ σv (G) such that k [ Ti . C⊂ i=1
Let M = V \ C and take Ci = C ∩ Ti and Mi = M ∩ Ti for all i = 1, . . . , k. Since the graph G does not have isolated vertices, then for all v ∈ M there exists an edge e of G with e = {v, v ′ }. Now, as C = V (G) \ M and C is a vertex cover we have that v ′ ∈ C, that is k [ (M ∩ N (Ci )). (1) M= i=1
Since Si = V (G) \ Ti = (C \ Ci ) ∪ (M \ Mi ) is a minimal vertex cover with |Si | ≤ n − σv (G) for all i = 1, . . . , k, then |C \ Ci | + |M \ Mi | = |(C \ Ci ) ∪ (M \ Mi )| = |Si | ≤ n − σv (G). Hence as M ∩ N (Ci ) ⊆ M \ Mi we have that |M ∩ N (Ci )| ≤ |M \ Mi | ≤ n − σv (G) − |C \ Ci | = |C| + α(G) − σv (G) − |C \ Ci | = |Ci | + α(G) − σv (G).
Taking
Ai = Ci \ (
i−1 [
j=1
Cj ) and Bi = (M ∩ N (Ci )) \
i−1 [
(2)
(M ∩ N (Cj )),
j=1
we have that |Ci \ Ai | ≤ |M ∩ N (Ci \ Ai )|,
(3)
since if |Ci \ Ai | > |M ∩ N (Ci \ Ai )|, then C \ (Ci \ Ai ) ∪ (M ∩ N (Ci \ Ai )) would be a vertex cover of cardinality |C \ (Ci \ Ai )| + |M ∩ N (C \ Ai )| < |C|, a contradiction. To finish the proof, we use the inequalities (2) and (3) to conclude that S |Bi | = |(M ∩ N (Ci ))| − |(M ∩ N (Ci )) ∩ i−1 j=1 (M ∩ N (Cj ))| S = |(M ∩ N (Ci ))| − |M ∩ N (Ci ) ∩ N ( i−1 j=1 Cj ))| (2) S ≤ |Ci | + α(G) − σv (G) − |M ∩ N (Ci ∩ i−1 j=1 Cj ))| (3) ≤ |Ci | + α(G) − σv (G) − |Ci \ Ai | = |Ai | + α(G) − σv (G).
Therefore P P (1) S α(G) = | ki=1 (M ∩ N (Ci ))| = ki=1 |Bi | ≤ ki=1 (|Ai | + α(G) − σv (G)) ≤ |C| + τ (G)(α(G) − σv (G))
= τ (G)[1 + α(G) − σv (G)]. 3
2
Remark 2.2 If α(G) > σv (G), then we have that α(G) = τ (G)[1 + α(G) − σv (G)] if and only if G is formed by a clique Kτ (G) with each vertex of this clique being the center of a star K1,α(G)−σv (G)+1 . Furthermore, if α(G) = σv (G) and α(G) = τ (G), then the graph has a perfect matching.
K1,α(G)−σv (G)+1 Kτ (G)
Figure 1: The graph formed by a clique Kτ (G) with each vertex of this clique being the center of a star K1,α(G)−σv (G)+1 . Let αcore (G) =
stable \set
|Mi |=α(G)
Mi and τcore (G) =
vertex \cover |Ci |=τ (G)
Ci ,
be the intersection of all the maximum stable sets and of all the minimum vertex covers of G, respectively. We have that G is a B-graph if and only if τcore (G) = ∅. Similarly, we have that a graph is τ -critical if and only if αcore (G) = ∅. We define Bα∩τ = V (G) \ {αcore (G) ∪ τcore (G)}. Proposition 2.3 Let G be a graph, then V (G) = αcore (G) ⊔ τcore (G) ⊔ Bα∩τ , furthermore (i) N (αcore (G)) ⊆ τcore (G), (ii) G[αcore (G)] is a trivial graph, (iii) G[Bα∩τ ] is both a τ -critical graph as well as a B-graph without isolated vertices. Proof. Clearly αcore (G)∩τcore (G) = ∅. Now, since G[V (G)\τcore (G)] is a B-graph, we have that, αcore (G) ⊂ V (G)\τcore (G) is the set of isolated vertices of G[V (G)\τcore (G)]. Therefore N (αcore (G)) ⊆ τcore (G), proving (i). Hence, we have that G[αcore (G)] is a graph without edges, giving (ii). Finally, by definition of Bα∩τ we obtain (iii). 2 Remark 2.4 It is easy to see that if v is an isolated vertex, then v ∈ αcore (G), note that in a similar way we have that if deg(v) > τ (G), then v does not belong to any stable set with α(G) vertices and therefore v ∈ τcore (G). Note that in general the induced graph G[Bα∩τ ] is not necessarily connected. 4
Example 2.5 To illustrate the previous result consider the following graph:
v1
v5
v3
v6
v7
v4
v2 we have that α(G) = 3, τ (G) = 4 and {v3 , v4 , v5 }, {v3 , v4 , v6 }, {v3 , v4 , v7 } are the maximum stable sets of G, then • αcore (G) = {v3 , v4 }, • τcore (G)] = {v1 , v2 }, • Bα∩τ = {v5 , v6 , v7 }. Corollary 2.6 Let G be a graph, then α(G) − |αcore (G)| ≤ τ (G) − |τcore (G)|. Proof. By Proposition 2.3 we have that G[Bα∩τ ] is a B-graph. Now, since α(G[Bα∩τ ]) = α(G) − |αcore (G)| and τ (G[Bα∩τ ]) = τ (G) − |τcore (G)|, and by applying Theorem 2.1 to G[Bα∩τ ] we obtain that α(G) − |αcore (G)| ≤ τ (G) − |τcore (G)|.
2
Corollary 2.7 ([1, Proposition 7]) If G is a B-graph without isolated vertices, then G is a τ -critical graph. Proof. Recall that a graph is a B-graph if and only if τcore (G) = ∅. Now, by Proposition 2.3(i) we have that if G is a B-graph, then N (αcore (G)) = ∅. Since G has no isolated vertices, then αcore (G) = ∅ and therefore G is a τ -critical graph. 2 Remark 2.8 The bound of Corollary 2.6 improves the bound given in [7, Theorem 2.11] for the number of vertices in αcore (G). Their result states: If G is a graph of order n and α(G) > (n + k − min{1, |N (αcore (G))|})/2, for some k ≥ 1, then |αcore (G)| ≥ k + 1. Moreover, if (n + k − min{1, |N (αcore (G))|})/2 is even, then |αcore (G)| ≥ k + 2. Notice that if α(G) ≥ n/2 + k′ /2, our bound gives,
|αcore (G)| ≥ k′ + |τcore (G)|.
5
2.1
An application to perfect hypergraphs
An hypergraph H is a pair (V, E) such that V ∩ E = ∅ and E is a subset of the set of all subsets of a finite set V . The maximum cardinality of an edge of H is denoted by rmax (H). An hypergraph H is called conformal if for each subset of vertices U of H we have that if each pair of vertices in U is contained in some edge of H, then U is contained in some edge of H. So H is conformal if and only if there exits a graph G on the same vertex set as H such that E consists of the inclusionwise maximal stable sets of G. The incidence matrix of the hypergraph H is the matrix M given by 1 if v ∈ E ME,v = 0 if v ∈ /E S A hypergraph H is called perfect if E∈E E = V and all vertices of the polyhedron QH = {x ∈ R|V | | x ≥ 0, M x ≤ 1}
are integers. We can assume without loss of generality that
T
Corollary 2.9 Let H be a perfect hypergraph with
2rmax (H) ≤ |V |.
E∈E
T
E = ∅.
E∈E
E = ∅, then
Proof. By [10, Theorem 82.3] we know that if H is a perfect hypergraph, then it is conformal. LetTG the graph such that the maximal stable sets S of G are the edges of H. We have that E∈E E = ∅ if and only if αcore (G) = ∅ and E∈E E = V if and only if τcore (G) = ∅. Therefore by Corollary 2.6 we have that 2rmax (H) = 2α(G) ≤ α(G) + τ (G) = |V |. 2
2.2
Conjectures
Definition 2.10 We define the ωe -clique covering number of G, denoted by ωe (G), as the greatest natural number m so that every edge in G belongs to a clique of size at least m. We also define σe (G) as ωe (G) Conjecture 2.11 Let G be a B-graph without isolated vertices, then ωe (G)σv (G) ≤ |V |. Furthermore, for all maximum stable sets M there exits disjoint sets Aj ⊂ V for j = 1, . . . , |M |, such that 6
i) |M ∩ Aj | = 1 for all j = 1, . . . , |M |, ii) G[Aj ] is a clique of order ωe (G) for all i = 1, . . . , |M |. The next conjecture was stated in [11, Conjecture 3.2.12]. Conjecture 2.12 Let H be a hypergraph with |V | = n, without isolated vertices and with |h| = r ∀h ∈ E. If σv (H) = α(H), then rσv (H) ≤ n. The last conjecture follows from Conjecture 2.11 by the following argument. Let H be an hypergraph and considerer the graph G(H) on the same vertex set of H in which v1 , v2 ∈ G(H) are adjacent if and only if they are adjacent in H. Clearly G(H) has the same number of vertices and the same stability number as H. Also observe that the clique number of G(H) is equal to the greatest cardinality of an edge in H. Moreover G(H) and H have the same σv and ωe covering numbers. Now, applying Conjecture 2.11 to the graph G(H) we have that, rσv (H) = ωe (G(H))σv (G(H)) ≤ n. Note that a graph is a B-graph if and only if σv (G) = α(G). The previous invariants of a graph (or hypergraph) satisfy that ωe (G) ≤ ωv (G) ≤ ω(G) and σe (G) ≤ σv (G) ≤ α(G). Observe that we cannot weaken the hypothesis in Conjecture 2.11, for example we cannot change the invariant ωe (G) by the invariant ωv (G) in the formula of 2.11. To see this consider the following graph:
v10
v5 v1
v2
v4
v3 v9
v6 v7
v8
We have that: • ωe (G) = 2 because the edge {v5 , v6 } is not in a K3 , • ωv (G) = 3 because {vi−1 , vi , v10−2i } for i = 1, 2, 3 and {v1 , v2 , v5 }, {v1 , v4 , v7 }, {v3 , v4 , v9 } are cliques, • ω(G) = 4 because and {v1 , v2 , v3 , v4 } is a clique, and 7
• σe (G) = α(G) = 4 because {v1 , v6 , v8 , v9 }, {v2 , v6 , v7 , v9 } {v3 , v5 , v7 , v10 } and {v4 , v5 , v8 , v10 } are stable sets. However ωv (G)α(G) = (3)(4) > 10 = n. A weaker version of Conjecture 2.11 is given in [4] and [5]. In these papers they proved that if G is a graph with n vertices such that every vertex belongs to a clique √ of cardinality q + 1 and an stable set of cardinality p + 1, then n ≥ p + q + 4pq. The following conjecture differs from the ones given above, in that it stresses the symmetry of the formulas with respect to the complement of a graph. Conjecture 2.13 Let G be a graph with n vertices and without isolated vertices. If α(G) = σe (G) and ω(G) = ωe (G), then ωe (G)σe (G) ≤ n. Remark 2.14 The hypothesis in Conjectures 2.11 and 2.13 are necessary. To see this considerer the following graph:
v8
v7 v4 v1
v3 v2
v5
v6
For this graph we have, • α(G) = 4 because {v5 , v6 , v7 , v8 } is a stable set, • ωe (G) = 3 because all the edges are in a K3 , and • ω(G) = 4, σe (G) = 3 because G ∼ = G. However, ωe (G)σe (G) = (3)(3) > 8 = n.
3
Graphs with a minimal number of edges
We will give a lower bound for the number of edges of a graph G as a function of the stability number α(G), the covering number τ (G) and the number of connected components c(G) of G. This is an answer to an open question posed by Ore in his book [8] which is a variant for connected graphs of a celebrated theorem of Turan [9]. We say that a graph G is q-minimal if there is no graph G′ such that 8
i) α(G′ ) = α(G), ii) τ (G′ ) = τ (G), iii) c(G′ ) = c(G) and iv) |E(G′ )| < |E(G)|. Hence if G is q-minimal, then either α(G) < α(G − e) (note that α(G) < α(G − e) if and only if τ (G) > τ (G − e)) or c(G) < c(G − e) for all the edges e of G. That is, an edge of a q-minimal graph is either critical or a bridge. Therefore the blocks of a q-minimal graph are α-critical graphs. Here a graph is α-critical if α(G − e) = α(G) + 1 for all the edges e of G and is τ -critical graph if τ (G − v) = τ (G) − 1 for all the vertices v of G. In order to bound the number of edges we introduce the following numerical function. Let a and t be two natural numbers and let ( a ) X zi Γ(a, t) = min | z1 + · · · + za = a + t and zi ≥ 0 ∀ i = 1, . . . , a . 2 i=1
Lemma 3.1 Let a and t be natural numbers, then r r+1 i) Γ(a, t) = (a − s) +s where a + t = r(a) + s with 0 ≤ s < a. 2 2 ii) Γ(a, t) − Γ(a − 1, t) ≤ 0 for all a ≥ 2 and t ≥ 1.
Moreover we have equality if and only if 0 ≤ t ≤ a − 1.
iii) Γ(a, t) − Γ(a, t − 1) = 1 + ⌊ t−1 a ⌋ for all a ≥ 1 and t ≥ 2. Pk Pk Pk iv) i=1 Γ(ai , ti ) ≥ Γ( i=1 ai , i=1 ti ) for all ai ≥ 1 and ti ≥ 1. Moreover we have that
Γ(a1 , t1 ) + Γ(a2 , t2 ) = Γ(a1 + a2 , t1 + t2 ) if and only if ⌊ at11 ⌋ = ⌊ at22 ⌋. ⌉ ≥ 1 + ⌊ at ⌋ for all a ≥ 2 and t ≥ 1. v) ⌈ 2(a−1+Γ(a,t)) a+t
Moreover we have equality if and only if t 6= a − 1.
Proof. (i) The case for a = 1 is trivial. For a ≥ 2 we will use the next result. Claim 3.2 Let n, m ≥ 1 be natural numbers with n > m + 1, then n m n−1 m+1 + > + . 2 2 2 2
9
n−1 n = n − 1. − Proof. It follows easily, since 2 2
a Let a ≥ 2 and t ≥ 1 be fixed natural numbers, (z1 , . . . , za ) ∈ N such that P z a + t and let L(z1 , . . . , za ) = ai=1 i . Now, if 2
2 Pa
i=1 zi
=
{z1 , . . . , za } = 6 {r, . . . , r , r + 1, . . . , r + 1} {z } | {z } | a−s
s
where a + t = r(a) + s with 0 ≤ s < a, then there exist zi1 and zi2 with zi1 > zi2 + 1. Applying Claim 3.2 we have that L(z1 , . . . za ) > L(z1 , . . . , zi1 − 1, . . . , zi2 + 1, . . . , za ) ≥ Γ(a, t), and therefore we obtain the result. (ii) Let a + t = ar + s with r ≥ 1 and 0 ≤ s < a, then a + t − 1 = (a − 1)(r + l) + t where r + s − 1 = (a − 1)l + t with l ≥ 0 and 0 ≤ t < a − 1. Using part (i) and after some algebraic manipulations we obtain that 2(Γ(a, t) − Γ(a − 1, t)) = (r 2 − r) + (l2 − l)(a − 1) + 2lt. Therefore Γ(a, t) − Γ(a − 1, t) ≥ 0, since r, l, t ≥ 0 and s2 − s ≥ 0 for all s ≥ 0. Moreover we have that Γ(a, t) − Γ(a − 1, t) = 0 if and only if (1, 0, t) (1, 1, 0) (r, l, t) = (0, 0, t) (0, 1, 0)
The two first possibilities imply that 0 ≤ t < a − 1 and t = a − 1, respectively. The last two cases are not possible. (iii) Let a + t − 1 = ar + s with r ≥ 1 and 0 ≤ s < α(G), then ar + (s + 1) if 0 ≤ s < a − 1 a+t= a(r + 1) if s = a − 1 and by (i) we have that
Γ(a, t) − Γ(a, t − 1) =
=
r r+1 r r+1 (a − s − 1) + (s + 1) − (a − s) +s 2 2 2 2 r+1 r r+1 a − + (a − 1) 2 2 2
r+1 r . − = r = a+t−1 a 2 2 10
(iv) Follows directly from the definition of Γ(a, t). (v) Let a + t = ar + s with r ≥ 1 and 0 ≤ s < α(G) then, by (i) we have that r r + 1 2 a−1+(a−s) 2 +s l l m m 2 2(a−1+Γ(a,t)) 2(a−1)+(a−s)r(r−1)+s(r+1)r = = a+t a+t a+t =
l
2(a−1)+r(ar+s)−r(a+s) ar+s
m
=r+
l
2(a−1)−r(a+s) ar+s
m
≥r=
The last inequality holds because m l 2(a−1)−r(a+s) ≥ 0 ⇔ −(ar + s) < −ar + sr + 2(a − 1) ar+s
a+t a
⇔ 2 < s(r + 1) + 2a ⇔ s > 0 or a ≥ 2. 2
Theorem 3.3 Let G be a graph, then |E(G)| ≥ α(G) − c(G) + Γ(α(G), τ (G)). Proof. We will use induction on τ (G), the covering number of G. For τ (G) = 1 it is easy to see that the unique connected graphs with τ (G) = 1 are the stars K1,n (α(K1,n ) = n − 1) and the result follows, since |E(K1,n )| = n − 1 = (n − 1) − 1 + 1 = α(K1,n ) + c(K1,n ) + Γ(n − 1, 1). In the same way it is easy to see that the unique graphs G with α(G) = 1 are the complete graphs Kn (τ (Kn ) = n − 1). Now we have, n n |E(Kn )| = =1−1+ = α(Kn ) + c(Kn ) + Γ(1, n − 1), 2 2 and it follows that the family of complete graphs satisfies the result. Moreover the graphs of both families are q-minimal graphs. So we can assume that the result is true for τ (G) ≤ k > 1. P Let G with τ (G) = k + 1. Since |E(G)| = si=1 |E(Gi )|, Psbe a q-minimal graphP α(G) = i=1 α(Gi ) and τ (G) = si=1 τ (Gi ) where G1 , . . . , Gs are the connected components of G, it follows from Lemma 3.1(iv) that we can assume with out loss of generality that G is connected and α(G) ≥ 2. Let e be an edge of G and considerer the graph G′ = G−e. We have two possibilities τ (G) ′ τ (G ) = τ (G) − 1 That is, an edge of G is either a bridge or critical. Case 1 First assume that G has no bridges, that is, G is a α-critical graph. Let v be a vertex of G of maximum degree. Since any α-critical graph is τ -critical we have that 11
τ (G − v) = τ (G) − 1 and α(G − v) = α(G), moreover as the α-critical graphs are blocks we have that G − v is connected. Now, by induction hypothesis we have that |E(G − v)| ≥ α(G) − 1 + Γ(α(G), τ (G) − 1) Using the formula α(G−v)+τ (G−v)
X
deg(vi ) = 2|E(G − v)|
i=1
we conclude that there must exit a vertex v ′ ∈ V (G − v) with 2(α(G) − 1 + Γ(α(G), τ (G) − 1)) 2|E(G − v)| ≥ . deg(v ′ ) ≥ α(G − v) + τ (G − v) α(G) + τ (G) − 1 Now by Lemma 3.1(iii) and (v) we have that
|E(G)| = |E(G − v)| + deg(v) ≥ α(G) − 1 + Γ(α(G), τ (G) − 1) + deg(v ′ ) ≥ α(G) − 1 + Γ(α(G), τ (G)). So, if the graph G has an edge that is a bridge, we have that c(G′ ) = c(G) − 1 = 2.
Let G1 and G2 be the connected components of G − e. Case 2 Assume that τ (G1 ) > 0 or τ (G2 ) > 0, then τ (G1 ) ≤ k and τ (G2 ) ≤ k and by the induction hypothesis we have that |E(G1 )| ≥ α(G1 ) − 1 + Γ(α(G1 ), τ (G1 )), |E(G2 )| ≥ α(G2 ) − 1 + Γ(α(G2 ), τ (G2 )). Using the above formulas and Lemma 3.1(iv) we have that |E(G)| = |E(G1 )| + |E(G2 )| + 1 ≥ α(G1 ) − 1 + α(G2 ) − 1 + Γ(α(G1 ), τ (G1 )) + Γ(α(G2 ), τ (G2 )) + 1 = α(G) − 1 + Γ(α(G1 ), τ (G1 )) + Γ(α(G2 ), τ (G2 )) (iv)
≥ α(G) − 1 + Γ(α(G), τ (G))
Note that α(G) = α(G1 ) + α(G2 ) and τ (G) = τ (G1 ) + τ (G2 ). Case 3 Assume that there does not exist a bridge with the above conditions, that is, for all the bridges of G we have that τ (G1 ) = 0 or τ (G2 ) = 0. In this case we must have that G is equal to an α-critical graph G1 with a vertex of G1 being the center of a star K1,l . Moreover we have that τ (G) = τ (G1 ) and α(G) = l + α(G1 ) because G1 is vertex-critical and therefore each vertex lies in a minimum vertex cover. Now using Case 1 and Lemma 3.1(ii), we obtain, E(G) = l + E(G1 ) ≥ l + (α(G1 ) − 1 + Γ(α(G1 ), τ (G1 ))) = α(G) − 1 + Γ(α(G1 ), τ (G)) (ii)
≥ α(G) − 1 + Γ(α(G), τ (G)).
2
Remark 3.4 After this paper was submitted, the authors learned that this result was also obtained independently in [2]. 12
References [1] Claude Berge, Some common properties for regularizable graphs, edge-critical graphs and B-graphs. Annals of Discrete Mathematics 12 (1982) 31–44. 2, 5 [2] Julie Christophe, Sophie Dewez, Jean-Paul Doignon, Sourour Elloumi, Gilles Fasbender, Philippe Gr´egoire, David Huygens, Martine Labb´e, Hadrien M´elot and Hande Yaman, Linear Inequalities among Graph Invariants: using GraPHedron to uncover optimal relationships. Manuscript, August 26, 2005. 12 [3] I. Gitler, Carlos E. Valencia, Bounds for invariants of edge-rings, Communications in Algebra, 33 (2005) 1603–1616. 2 [4] Fred Galvin, Another note on cliques and independent sets, Journal of Graph Theory 35, 3, (2000), 173–175. 8 [5] Roger C. Entringer, Wayne Goddard, Michael A. Henning, A note on cliques and independent sets. Journal of Graph Theory 24, 1, (1997), 21–23. 8 [6] J. Harant and I. Schiermeyer, On the independence number of a graph in terms of order and size, Discrete Math. 232 (2001), 131–138. [7] Vadim E. Levit, Eugen Mandrescu, Combinatorial properties of the family of maximum stable sets of a graph, Discrete Applied Math. 117 (2002), 149–161. 5 [8] O. Ore, Theory of graphs, American Mathematical Society Colloquium Publications, Vol. XXXVIII, American Mathematical Society, Providence, R. I., 1962. 8 [9] P. Tur´ an, Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok 48 (1941), 436–452. 8 [10] A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics 24 (2003) Springer-Verlag, Berlin. 6 [11] Carlos E. Valencia, Studies on graded rings associated to graphs, Ph.D. Thesis, (2003) CINVESTAV-IPN. 7
13
