Cubic graphs with small independence ratio

Let $i(r,g)$ denote the infimum of the ratio $\frac{\alpha(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $\alpha(G)$ is the independence number of $G$, and let $i(r,\infty) := \lim\limits_{g \to \infty} i(r,g)$. Recently, several new lower bounds of $i(3,\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \infty) \ge 0.4375$, and Cs\'oka improved it to $i(3,\infty) \ge 0.44533$ in 2016. Bollob\'as proved the upper bound $i(3,\infty)<\frac{6}{13}$ in 1981, and McKay improved it to $i(3,\infty)<0.45537$ in 1987. There were no improvements since then. In this paper, we improve the upper bound to $i(3,\infty) \le 0.454.$


Introduction
A set S of vertices in a graph G is independent if no two vertices of S are joined by an edge. The independence number, α(G), is the maximum size of an independent set in G. The independence ratio, i(G), of a graph G is the ratio α(G) |V (G)| . For positive integers r and g, i(r, g) denotes the infimum of i(G) over the r-regular graphs of girth at least g, and i(r, ∞) denotes lim g→∞ i(r, g). The first interesting upper bounds on i(r, ∞) were obtained by Bollobás [2] in 1981. In particular, he proved i(3, ∞) < 6 13 . Refining the method, McKay [11] in 1987 showed Theorem 1 (McKay [11]). i(3, ∞) < 0.45537.
In the next 30 years, there were no improvements of Theorem 1, but recently some interesting lower bounds on i(r, ∞) and in particular on i(3, ∞) were proved. Hoppen [6] showed i(3, ∞) ≥ 0.4328. Then Kardoś, Král and Volec [10] improved the bound to 0.4352. Csóka, Gerencsér, Harangi, and Virág [5] pushed the bound to 0.4361 and Hoppen and Wormald [7] -to 0.4375. Moreover, Csóka et al [5] claimed a computer assisted lower bound i(3, ∞) ≥ 0.438, and Csóka [4] later improved the bound to 0.44533. Our result is an improvement of (1) to i(3, ∞) ≤ 0.454. The improvement is small, but it decreases the gap between the upper and lower bounds on i(3, ∞) by approximately 14%. The proof uses the language of configurations introduced by Bollobás [3], and shows that "many" 3-regular configurations have "small" independence ratio. The proof of our improvement is based on analyzing the presence not of largest independent sets, but of larger structures, so called MAI-sets (defined in Section 3) that contain largest independent sets.

Notation
We mostly use standard notation. The complete n-vertex graph is denoted by K n . If G is a multigraph and v, u ∈ V (G), then E G (v, u) denotes the set of all edges in G connecting v and u, e G (v, u) |E G (v, u)|, and deg G (v) u∈V (G)\{v} e G (v, u). By ∆(G) we denote the maximum degree of G, and by g(G) -the girth (the length of a shortest cycle) of G.

The Configuration Model
The configuration model in different versions is due to Bender and Canfield [1] and Bollobás [3]. Our work is based on the version of Bollobás. Let n be an even positive integer and V n = [n]. Consider the Cartesian product W n = V n × [3]. A configuration/pairing (of order n and degree 3) is a perfect matching on the vertex set W n . There are (3n − 1) · (3n − 3) · . . . · 1 = (3n − 1)!! such matchings.
Let F 3 (n) denote the collection of all (3n−1)!! possible pairings on W n . We project each pairing F ∈ F 3 (n) to a multigraph π(F ) on the vertex set V n by ignoring the second coordinate. Then π(F ) is a 3-regular multigraph (which may or may not contain loops and/or multiple edges). Let π(F 3 (n)) = {π(F ) : F ∈ F 3 (n)} be the set of 3-regular multigraphs on V n . By definition, each simple graph G ∈ π(F 3 (n)) corresponds to (3!) n distinct pairings in F 3 (n). (2) We will call the elements of V n -vertices, and of W n -points.
be the set of all cubic graphs with vertex set V n = [n] and girth at least g and G g (n) = {F ∈ F 3 (n) : π(F ) ∈ G g (n)}.
We will heavily use the following result: Theorem 4 (Wormald [13], Bollobás [3]). For each fixed g ≥ 3, Remark. When we say that a pairing F has a multigraph property A, we mean that π(F ) has property A.
Since dealing with pairings is simpler than working with labeled simple regular graphs, we need the following well-known consequence of Theorem 4. Corollary 5 ([11](Corollary 1.1), [9](Theorem 9.5)). For fixed g ≥ 3, any property that holds for π(F ) for almost all pairings F ∈ F 3 (n) also holds for almost all graphs in G g (n). Definition 6. For a graph G, let I(G) denote the total number of all independent sets in G, including the empty set. For all integer r ≥ 0, g ≥ 3, we define I(r, g) = inf I(G) 1/|V (G)| , where the infimum is over all graphs G of maximum degree at most r and girth at least g.
Recall that the Fibonacci numbers F n are defined by F 1 = F 2 = 1, and Lemma 7 (McKay [11]). For any g ≥ 4, Remark 8. The numbers s − 1 and s + 1 in Lemma 7 are even. Therefore, Since the function (1 − ϕ −2s ) 1/s monotonically increases for s ≥ 1, and ϕ(1 − ϕ −18 ) 1/9 ≥ 1.618002, we conclude that for each graph H with maximum degree at most 2 and girth at least 8, (a) = 1, then without loss of generality, we may assume a ∈ A − A . Then b has no neighbors in A , and A ∪ {b} is an independent set in G with size |A | + 1, again contradicting the definition of α(G).

MAI sets in cubic graphs
Let A be a MAI set in G ∈ G 16 (n). Denote the set of vertices with degree 1 in G[A] by Y , the set of vertices with degree 1 in G[B] by Z. We introduce notation for the sizes of the sets: Let x := |A |, s := |Y |/2, t := |Z|/2, and i := n 2 − |A|. Then |A| = n 2 − i and |B| = n 2 + i. Lemma 12. i ≥ 0 and t ≥ s.
Proof. We count the number of edges with one end in A and one end in B in two ways. We have i.e., t − s = 3i.

Lemma 13.
If G ∈ G 5 (n), then (i) each vertex in Z has degree at most one to Y ; (ii) each vertex in Y has degree at most one to Z.
Proof. (i) Suppose z ∈ Z and N G (z) = {z , y 1 , y 2 }, where z ∈ Z and y 1 , y 2 ∈ Y . Since g(G) ≥ 4, y 1 y 2 , y 1 y 2 E(G), and so A − y 1 − y 2 contains an independent set A with |A | = α(G). Thus the set A + z is an independent set of size α(G) + 1 contradicting the definition of α(G).
(ii) Similarly, suppose y ∈ Y and N G (y) = {y , z 1 , z 2 }, where y ∈ Y and z 1 , z 2 ∈ Z. Then A − y contains an independent set A with |A | = α(G).
By Part (i), a 1 , a 2 Y . Since g(G) ≥ 5, a 2 a 1 . Then (A − y) ∪ {z 1 , z 2 } is an AI set containing A and is larger than A, a contradiction.
Let J = {y 1 z 1 , . . . , y j z j } be the set of all edges connecting Y with Z in G. By Lemma 13, J is a matching in G. Define an auxiliary graph H = H(A) as follows: V (H) = J, and y z is adjacent to y z if y y ∈ E(G) or z z ∈ E(G). By construction, the maximum degree of H is at most 2 and a cycle of length c in H corresponds to a cycle of length 2c in G.
If a vertex a ∈ A − Y is adjacent to two vertices, say z 1 , z 2 in Z 1 , then the set (A − a) ∪ {z 1 , z 2 } is independent and is larger than A , a contradiction. Thus, A 1 is an AI set. Since |A 1 | = |A|, this proves the lemma.

Restating the theorem
We will use Theorem 1 of McKay in the following stronger form.
Theorem 16 (McKay [11]). For every ε > 0, there exists an N > 0 such that for each n > N , We will show that "almost all" cubic labeled graphs of girth at least 16 have independence ratio at most 0.454. In view of Theorem 4, the following more technical statement implies Theorem 2.
Theorem 17. For every ε > 0, there is an N > 0 such that for each n > N, The rest of the paper is a proof of Theorem 17. By definition, every graph has a MAI set. So, for large n, nonnegative integers x ≥ 0.454n and i ≤ n 2 − x, and each set A of size n 2 − i with a fixed matching of size n 2 − i − x we will estimate the total x-weight of configurations F ∈ G 16 (n) in which A forms a MAI set. The idea of the weight (used by McKay in [11]) is to decrease overcount of the configurations containing a given MAI set, but guarantee that the total weight of each configuration containing at least one MAI set with independence number x would be at least 1.

Setup of the proof of Theorem 17
andR is a matching on a subset ofÂ with |R| = |R| such that for each edge (i, j)(i , j ) ∈R, ii ∈ R. In other words, each edge e ∈ R is obtained from an edge inê ∈R by ignoring the second coordinates of the ends ofê, and this mapping is one-to-one.
Lemma 18. Let n be a positive even integer and x be an integer with 0.454n < x ≤ 0.45537n. The number of pairings F ∈ G 16 (n) such that π(F ) has a MAI set A with |A | = x is at most Proof. By (10), it is enough to show that σ(n, x, 16) ≤ q(x, n). Below we describe a procedure of constructing for every AI-pair (A, R) on [n] with α(A, R) = x all pairings in F ∈ G 16 (n) for which A is a MAI set. Not every obtained pairing will be in G 16 (n) and some pairings will have independence number larger than x, but every F ∈ G 16 (n) such that A is a MAI set in π(F ) will be a result of this procedure.
ways to do it. Then there are 3 n−2x−2i ways to decide which point of each chosen end of an edge in R will be the end of the corresponding edge in F .

Similarly to
Step 2, we have After that there are 3 n−2x+4i ways to decide which point of each chosen end of an edge in R will be the end of the corresponding edge in F .  (8), 6. Now we choose for each remaining free point p from vertices in Y a free point q in a vertex in B − Z and add edge pq. There are 7. Similarly to Step 6, we choose for each remaining free point q from vertices in Z a free point p in a vertex in A − Y and add edge pq. There are In the proofs below we will use Stirling's formula: For every n ≥ 1, We will also use the notation ∂ ∂j to denote the partial derivative with respect to j. Moreover, we use the domain x ≥ 0 and define ln(0) = −∞ when we consider ln x.

Lemma 19. Let n be a positive even integer and x be an integer satisfying
Then Proof. We write q(x, n) as a double sum of i and j and let r(x, n, i, j) be the function inside the double sum of q(x, n), i.e., r(x, n, i, j).
Then certainly, So, it is enough to estimate r(x, n, i, j). We know that Recall that Introducing new variables χ := x n , ζ := i n , and ξ := j n and using Stirling's formula (11) Therefore, This proves the lemma.
Recall that the domain of h(χ, ζ, ξ) is Ω defined in (12). Our main goal now is to show that We do this in the next section, and then Theorem 17 easily follows.
Thus which implies that The proof of the next lemma is similar but significantly simpler. It is mostly a routine bounding some expressions. So, we present the proof of Lemma 21 in Appendix 2.

Completion of the proof of Theorem 17
By (14) and Lemma 19, for all positive integers n and x such that n is even and 0.454n It follows that