A Set and Collection Lemma

The neighborhood of a vertex v ∈ V is the set N (v) = {w : w ∈ V and vw ∈ E}, while the neighborhood of

A set S ⊆ V (G) is independent (stable) if no two vertices from S are adjacent, and by Ind(G) we mean the set of all the independent sets of G. An independent set of maximum cardinality will be referred to as a maximum independent set of G, and the independence number of G is α(G) = max{|S| : S ∈ Ind(G)}.

A matching (i.e., a set of non-incident edges of G) [4,13].

Let Ω(G) denote the family of all maximum independent sets of G and core(G) = ∩{S : S ∈ Ω(G)} [10], while corona(G) = ∪{S : S ∈ Ω(G)} [3].

In this paper we introduce the “Set and Collection Lemma”. It is both a generalization and strengthening of a number of elegant observations including the “Maximum Stable Set Lemma” due to Berge and the “Clique Collection Lemma” due to Hajnal.

It is clear that the statement “there exists a matching from a set A into a set B” is stronger than just saying that |A| ≤ |B|. The “Matching Lemma” offers both a powerful tool validating existence of matchings and its most important corresponding inequalities, emphasized in the “Set and Collection Lemma” and its corollaries.

Then the following assertions are true:

(i) there exists a matching from S -∩Λ into ∪Λ -S;

(ii) there is a matching from S -X into X -S;

(iii) there exists a matching from S ∩ X -∩Λ into ∪Λ -(X ∪ S).

Proof. Let B 1 = ∩Λ and B 2 = ∪Λ.

(i) In order to prove that there is a matching from S -B 1 into B 2 -S, we use Hall’s Theorem, i.e., we show that for every A ⊆ S -B 1 we must have

Assume, in a way of contradiction, that Hall’s condition is not satisfied. Let us choose a minimal subset à ⊆ S -B 1 , for which à > N à ∩ B 2 .

There exists some W ∈ Λ such that à W , because à ⊆ S -B 1 . Further, the inequality à ∩ W < à and the inclusion

because we have selected à as a minimal subset satisfying à > N à ∩ B 2 . Therefore,

is an independent set of size greater than |W | = α (G), which is a contradiction that proves the claim.

(ii) It follows from part (i) for Λ = {X}.

(iii) By part (i), there exists a matching from S -∩Λ into ∪Λ -S, while by part (ii), there is a matching from S -X into X -S. Since X is independent, there are no edges between

Therefore, there exists a matching

For example, let us consider the graph G from Figure 1 and

In addition, we have

The assertions of Matching Lemma may be false, if the family Λ is not included in Ω (G). For instance, if

Proof. Let X ∈ Λ. By Matching Lemma (iii), there is a matching from S ∩ X -∩Λ into ∪Λ -(X ∪ S). Hence we infer that

Therefore, we obtain that

Proof. Let S ∈ Λ. By Set and Collection Lemma, we get that

If Λ = Ω(G), then Corollary 2.3 gives the following.

Corollary 2.4 For every graph G, it is true that Proof. Notice that for every S ∈ Ω (G), we have core(G) This contradiction proves that the inequality

In other words, the bound in Proposition 2.5 is tight. It has been shown in [11] that

is satisfied by every König-Egerváry graph G, and taking into account that clearly

we infer that the König-Egerváry graphs enjoy the following nice property.

It is worth mentioning that the converse of Proposition 2.7 is not true. For instance, see the graph G 2 from Figure 3, which has α

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. Applying Matching Lemma (i) to Λ = Ω(G) we immediately obtain the following.

Corollary 2.9 [3] For every S ∈ Ω(G), there is a matching from Score(G) into corona(G) -S.

Since every maximum clique of G is a maximum independent set of G, Corollary 2.3 is equivalent to the “Clique Collection Lemma” due to Hajnal.

Another application of Matching Lemma is the “Maximum Stable Set Lemma” due to Berge.

Corollary 2.11 [1], [2] An independent set X is maximum if and only if every independent set S disjoint from X can be matched into X.

Proof. Matching Lemma (ii) is, essentially, the “if " part of corollary.

For the “only if " part we proceed as follows. According to the hypothesis, there is a matching from S -X = S -S ∩ X into X, in fact, into X -S ∩ X, for each S ∈ Ω (G) -{X}. Hence, we obtain

which clearly implies X ∈ Ω (G).

In this paper we have proved the “Set and Collection Lemma”, which has been crucial in order to obtain a number of alternative proofs and/or strengthenings of some known results. Our main motivation has been the “Clique Collection Lemma” due to Hajnal [7]. Not only this lemma is beautiful but it is in continuous use as well. Let us only mention its two recent applications in [8,12]. Proposition 2.7 claims that 2 • α(G) = |c

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