Stability for maximal independent sets

Answering questions of Y. Rabinovich, we prove"stability"versions of upper bounds on maximal independent set counts in graphs under various restrictions. Roughly these say that being close to the maximum implies existence of a large induced matching or triangle matching (depending on assumptions).


Introduction
Denote the number of maximal independent sets in a graph G by mis(G). We recall two well-known bounds for these numbers: Theorem 1 (Moon-Moser [10]). For any n-vertex graph G, mis(G) 3 n/3 , with equality iff G is the disjoint union of n/3 triangles.
As usual M is an induced matching of G if it is an induced subgraph of G that is a matching. Similarly, T is an induced triangle matching of G if it is an induced subgraph of G that is a vertex disjoint union of triangles.
Write itm(G) for the number of triangles in a largest induced triangle matching in G, and im(G) for the number of edges in a largest induced matching.
In what follows we will usually prefer to work with log mis (log = log 2 ), thought of as the number of bits needed to specify a maximal independent set. Note that itm(G) log 3 and im(G) are obvious lower bounds on log mis(G). We will be interested in questions suggested to us a few years ago by Yuri Rabinovich [11] concerning "stability" aspects of upper bounds on mis, meaning, roughly: does large mis imply existence of a large induced triangle matching or large induced matching (as appropriate)? Formally, his conjectures were unquantified versions of the following three statements, whose proofs are the content of the present work. (The questions were motivated by [12], which includes a proof of Theorem 4 for bipartite graphs.) Theorem 3. For any ε > 0, there is a δ = δ(ε) = Ω(ε) such that for an n-vertex graph G, if itm(G) < (1 − ε) n 3 then log mis(G) < ( 1 3 log 3 − δ)n. Theorem 4. For any ε > 0, there is a δ = δ(ε) = Ω(ε) such that for a triangle-free n-vertex graph G, if im(G) < (1 − ε) n 2 then log mis(G) < ( 1 2 − δ)n. One reason to be interested in Theorem 4-or in what its proof actually gives; see Theorem 14 below-is its key role in a proof of the following statement, which was conjectured in [6] (see also [2]) and whose proof is completed in [7] and [8].
Theorem 5. With Q n denoting the n-dimensional Hamming cube, While Theorem 14 is one of the easier ingredients in the proof of Theorem 5, it is in some sense the basis for the whole; in particular, it was understanding the connection between induced matchings and stability that first suggested that the conjecture of [6], which had seemed out of reach, might in fact be manageable.
Theorem 4 applies to bipartite graphs, of course. If G is bipartite with bipartition X ∪ Y , then log mis(G) is trivially at most min{|X|, |Y |} (since a maximal independent set is determined by its intersection with either of X, Y ); so the statement is uninteresting unless G is close to balanced. But Rabinovich asked whether something analogous also holds for unbalanced (bipartite) G; more precisely, whether something along the following lines is true.
The proof of this is easily adapted to |Y | = Bn (with δ then δ(ε, B)), but to keep things simple we just state the result for B = 2.
Rabinovich suspected that, as in Theorems 3 and 4, δ(ε) should be linear in ε, but this is not true. In fact, Theorem 6 is tight (up to the implied constant); a construction to show this will be given in Section 4.2.
The rest of the paper is organized as follows. Section 2 recalls some background, in particular Füredi's upper bounds on mis for paths and cycles, and Shearer's entropy lemma. Section 3 gives the proofs of Theorems 3 and 4. The proof of Theorem 6 and the example to show its tightness are given in Section 4.
The proofs of Theorems 3 and 4 are similar, while that of Theorem 6 is related but somewhat trickier. The general approach has its roots in an idea for counting (ordinary) independent sets due to A.A. Sapozhenko [13], [14].
Strictly speaking we prove the theorems only for sufficiently large n, since we occasionally hide minor terms in o(1)'s. Of course combined with the characterizations of equality in Theorems 1 and 2 this does give the stated versions, though the δ's we produce may not be valid for small n. Since we are really interested in large n anyway, this approach seems preferable to carrying explicit error terms.
Notation. We use "∼" for adjacency, N (x) for the neighborhood of x, N (S) = ∪ x∈S N (x), and d S (x) = |N (x) ∩ S|. As usual, G[S] is the subgraph of G induced by S ⊆ V (G).

Preliminaries
For the proof of Theorem 4 we need the following upper bounds on mis for paths and cycles, given by Z. Füredi [4]. 1. For P n , the path with n vertices, mis(P n ) 2γ n−2 .
2. For C n , the cycle with n vertices, mis(C n ) 3γ n−3 .
We very briefly recall a few entropy basics (see also e.g. [9]). For discrete random variables X, Y , the (binary) entropy of X is and conditional entorpy of X given Y is x p(x|y) log 1 p(x|y) (where p(x) = P(X = x) and p(x|y) = P(X = x|Y = y)).
(a) H(X) log |Range(X)|, with equality iff X is uniform from its range; In addition to these very basic properties we need the following version of Shearer's Lemma [1]. then Finally, we will need the following standard fact (see e.g. Lemma 16.19 in [3]; this is also implied by Lemma 9 with α A equal to 1 if |A| = 1 and zero otherwise).

Proofs of Theorems and 4
In this section, I is always a maximal independent set in G.
The basis for what we do is the following algorithm, which, given G and I, encodes some portion of Ias a string ξ(I), with the numbers of possibilities for both ξ(I) and the full specification of I given ξ(I) not too large.

Proof of Theorem 3
The argument for Theorem 3 goes roughly as follows. Noting that ξ(I) determines both X * and I \ X * , and we find that mis(G) (If we restrict the sum to possible ξ's-those corresponding to actual I's-then we have equality in (5).) It turns out that running the algorithm for very long is "expensive" in the sense that the loss in |X * |, and so in possibilities for I ∩ X * , outweighs what is contributed to (5) by possibilities for ξ; this limits the number of I's with t(I) large. Similarly, the difference between the bounds in Theorems 1 and 2 says there are "few" I's for which the triangle-free part of G * is large. (Note G * , having maximum degree at most two, is a disjoint union of triangles and a triangle-free part, below called R.) But the part of mis(G) corresponding to I's for which both t and R are small must come mainly from counting choices for the restriction of I to the triangles of G * , and these are limited by our assumption on itm(G).
To begin with, the following lemma bounds the number of I's with large t(I).
the electronic journal of combinatorics 27(1) (2020), #P1.59 Let T = T (I) be the union of the triangles in X * (so the unique maximal induced triangle matching in G * ), R = R(I) = G * [X * \ V (T )], and r = r(I) = |V (R)|. Note that there are no edges between V (T ) and V (R), since G * has maximum degree at most 2, so mis(G * ) = mis(T )mis(R). (8) Note also that R is triangle-free, so log mis(R) r/2 by Theorem 2. Now, the following lemma bounds the number of I's with large r.

Proof of Theorem 4
We first give the slightly stronger version of Theorem 4 mentioned in Section 1. Theorem 14. For any ε > 0 there is a δ = Ω(ε) such that for any n-vertex, triangle free G, log mis(G, ε) < (1 − δ)n/2.
(We have omitted the corresponding strengthening of Theorem 3.) As mentioned earlier, the argument for Theorem 14 is similar to the one in Section 3.2, so we will try to be brief. We again start from the algorithm in Section 3.1, and continue to use the notation (X * , G * etc.) defined in the paragraph following the algorithm's description. (For most of this we just need I ∈ I; the role of I ε will appear in Lemma 18.) Proof. Arguing as for (7) in Section 3.2, we obtain |{I ∈ I : t(I) = t, s(I) = s}| t s 2 (n−(t+3s))/2 , where we used mis(G * ) 2 (n−(t+3s))/2 , as given by Theorem 2 (since G is triangle-free). Thus Note that R is triangle-free, so is a vertex-disjoint union of isolated vertices, cycles with at least 4 vertices, and paths with at least 3 vertices. Combining this with Proposition 7, we obtain an upper bound for mis(R). (Recall that γ ≈ 1.325 was defined in Proposition 7.) Lemma 16. With R and r as above, mis(R) (3γ) r/4 . Proof. Let l p (resp. l c ) be the number of vertices in the union of all paths (resp. cycles) in R. Clearly l p + l c r, while the number of paths (resp. cycles) in R is at most l p /3 (resp. l c /4). Thus where the first inequality is given by Proposition 7 and the second follows from the fact that ( for any I ∈ I ε with r(I) = r. Therefore, |{I ∈ I ε : t(I) = t, r(I) = r}| 2 t 2 (1−ε)n/2 (3γ) r/4 , and summing over the relevant t's and r's gives |{I ∈ I ε : t(I) < xn, r(I) < yn}| 2 (1−ε)n/2 · 2 xn+1 · ((3γ) 1/4 − 1) −1 (3γ) (yn+1)/4 ; so we have (16).

Proof of Theorem 6
For a bipartite graph G on X ∪ Y , say X ⊆ X is irredundant if ∀x ∈ X , N (x) ⊆ N (X \ {x}). (So for this discussion "irredundant" sets are always subsets of X.) Denote the number of irredundant sets in G by irr(G).
Proof. This follows from the observation that for each maximal independent set I there is an irredundant set J ⊆ I ∩ X with N (J) = N (I ∩ X) (= Y \ I); namely, this is true whenever J ⊆ I ∩ X is minimal with N (J) = N (I ∩ X).
Thus the following statement implies Theorem 6.
For the rest of this section, G is as in Theorem 20.

Proof
The algorithm we use for Theorem 20 is slightly different from the one in section 3.1. In what follows, I is always an irredundant set (thus I ⊆ X).
Algorithm Let X 0 = X, Y 0 = Y and M = M ε = 12/ε. Fix an order "≺" on X. For a given I, repeat for i = 1, 2, . . .: 1. Let x i be the first vertex of X i−1 in ≺ among those with largest degree in X i−1 .
Consider a random (uniform) irredundant set I. Our various parameters (ξ, ψ, . . .) are then random variables, which will be denoted by ξ and so on. Since each of I and (ξ, ψ) determines the other and ξ determines t, we have (using parts for any ξ. Thus the sum of the last two terms in (17) is at most and we would like to somewhat improve these bounds. (Since we aim for H(I) < n−Ω(n), the log n in (17) is irrelevant.) The next lemma, giving such a gain in (18) when t is small, is our main point.
Lemma 21. For any ξ with |ξ| = t < εn/2, Proof. Given ξ as in the Lemma, set We have where the last inequality holds since for each x ∈ X * \X there is some y x ∈ Y * with N X * (y x ) = {x}, and {(x, y x ) : x ∈ X * \X} is an induced matching of G * of size |X * \X| = n − t − |X|. Thus |X| > εn − t > εn/2.
For each x ∈X fix some Z x ⊆ X * \ {x} such that and ∀z ∈ X * |{x ∈X : z ∈ Z x }| < 2M.
To see that we can do this: For each y ∈ N Y * (X) let Π y be a partition of N X * (y) into blocks of size 2 or 3. (Note y ∈ N Y * (X) implies d X * (y) 2.) Then to form Z x , for each y ∈ N Y * (x) choose one x = x from the block of Π y containing x and take x ∈ Z x . Note that each x ∈ X * has degree less than M in G * (see step 3 of the algorithm), so we have Let W x = Z x ∪ {x} (x ∈X), and ψ A = ψ ∩ A for any A ⊆ X. Note that for each x ∈ X * , (The first inequality follows from irredundancy: we cannot have ψ Wx = W x .) Now aiming to use Lemma 9, form α : 2 X * → R 0 by assigning weight 1/(2M ) to each W x (thus assigning each set weight some multiple of 1/(2M ), with the total weight of the sets containing any given x at most 1 by (22)) and supplementing with weights on the singletons to get to (1). Then by Lemma 9, Now (23) and the fact that α assigns total weight |X|/(2M ) to the W x 's give

Tightness
Define a bipartite graph B m on X ∪ Y = [m] ∪ [2m] (disjoint copies, of course) as follows.
1. If x ∈ X and x m − 1, then x ∼ y iff y = x or y = m − 1 + x.
It is easy to see that im(B m ) = m − 1, and mis(B m ) = 2 m − 1.