Classical Lovász Local Lemma Lovász Local Lemma (or LLL) is a very powerful tool in combinatorics and probability theory to show the possibility of avoiding all “bad” events under some “weakly dependent” condition, and has numerous applications. Formally, given a set of bad events in a probability space, LLL provides the condition under which . The dependency among events is usually characterized by the dependency graph. A dependency graph is an undirected graph such that for any vertex , is independent of , where stands for the set of neighbors of in . In this setting, finding the conditions under which is reduced to the following problem: given a graph , determine its abstract interior
which is the set of vectorssuch that for any event set with dependency graph and probability vector . Local solutions to this problem, including the first LLL proved in 1975 by Erdős and Lovász , are referred as abstract-LLL (or ALLL).
The most frequently used abstract-LLL is as follows:
Theorem 1.1 ().
Given a dependency graph and a probability vector , if there exist real numbers such that for any , then .
Shearer  provided the exact characterization of with the independence polynomial defined as follows.
Definition 1.1 (Multivariate independence polynomial).
Let , and let be the set of all independent sets of . Then we call the multivariate independence polynomial.
A probability vector is called above Shearer’s bound for a dependency graph if there is a vertex set such that for the corresponding induced subgraph . Otherwise we say is below Shearer’s bound.
The tight criterion under which abstract version LLL holds provided by Shearer is as follows.
Theorem 1.2 ().
For a dependency graph and probabilities the following conditions are equivalent:
is below Shearer’s bound for .
for any probability space and events having as dependency graph and satisfying , we have .
In other words, if and only if is below Shearer’s bound for .
Another important version of LLL, variable version Lovász Local Lemma (or VLLL), which exploits richer dependency structures of the events, has also been studied [27, 23]. In this setting, each event can be fully determined by some subset
of a set of mutually independent random variables. Thus, the dependency can be naturally characterized by the event-variable graph defined as follows. An event-variable graph is a bipartite graph such that for any , there is an edge . Similar to the abstract-LLL, the VLLL is for solving the following problem: given a bipartite graph , determine its variable interior which is the set of vectors such that for any variable-generated event system with event-variable graph and probability vector .
The VLLL is important because many problems in which LLL has applications naturally conform with the variable setting, including hypergraph coloring , satisfiability [15, 14], counting solutions to CNF formulas , acyclic edge coloring , etc. Moreover, most of recent progresses on the algorithmic aspects of LLL are based on the variable model [34, 36, 27].
A key problem around the VLLL is whether Shearer’s bound is tight for variable-LLL . Formally, given a bipartite graph , its base graph is defined as the graph such that for any two nodes , there is an edge if and only if and share some common neighbor in . That is to say, is a dependency graph of the variable-generated event system with event-variable graph . Thus, we have immediately. If , we say that Shearer’s bound is not tight for , or has a gap. The first example of gap existence is a bipartite graph whose base graph is a cycle of length 4 . Recently, He et al.  have shown that Shearer’s bound is generally not tight for variable-LLL.
Quantum Satisfiability and Quantum Lovasz Local Lemma Most systems of physical interest can be described by local Hamiltonians where each -local term acts nontrivially only on at most qudits. We say is frustration free if the ground state of is also the ground state of every . Let be the projection operator on the excited states of and , and it is easy to see the frustration freeness of and are the same. Henceforth, we only care about the Hamiltonians which are projectors. Determining whether a given is frustration free (or satisfiable, in computer science language), known as the quantum satisfiability problem, is a central pillar in quantum complexity theory, and has many applications in quantum many body physics.
Unfortunately, quantum satisfiability problem has been shown to be QMA-complete 
, which is widely believed to be intractable in general even for quantum computing. This makes it highly desirable to search for efficient heuristics and algorithms in order to, at least, partially answer this question.
In the seminal paper, by generalizing the notations of probability and independence as described in the following table, Ambainis et al.  introduced a quantum version LLL (or QLLL) respect to the dependency graph, i.e., a sufficient condition under which the Hamiltonian is guaranteed to be frustration free with given relative dimensions. Here, the relative dimension of a Hamiltonian is defined as that of the subspace it projects. With QLLL, they greatly improved the known critical density for random -QSAT from  to , almost meet the best known upper bound .
|Probability space||Vector space:|
Recently, Sattath et al.  generalized Shearer’s theorem to QLLL respect to the interaction bipartite graph, which can be viewed as the quantum analog of the classical event-variable graph, and showed that Shearer’s bound is still a sufficient condition here. Remarkably, the probability threshold of Shearer’s bound turns out to be the first negative fugacity of the hardcore lattice gas partition function, which has been extensively studied in classical statistical mechanics. Utilizing the tools in classical statistical mechanics, they concretely apply QLLL to evaluating the critical threshold for various regular lattices. In contrast to VLLL  which goes beyond Shearer’s bound generally, they conjectured that Shearer’s bound is tight for QLLL, which, if true, would have important physical significance and several striking consequence .
In the past few years, as a special case of quantum satisfiability problem, the commuting local Hamiltonian problem (CLH), where for all and , has attracted considerable attention [6, 1, 39, 19, 2]. Commuting Hamiltonians are somewhat “halfway” between classical and quantum, and capable of exhibiting intriguing multi-particle entanglement phenomena, such as the famous toric code . CLH interests people not only because the commutation restriction is natural and often made in physics, but also it may help us to understand the centrality of non-commutation in quantum mechanics. CLH can be viewed as a generalization of the classical SAT, thus CLH is at least NP-hard, and as a sufficient condition, the commuting version LLL (or CLLL) is desirable and would have various applications.
The QLLLs provide sufficient conditions for frustration freeness. A natural question is whether there is an efficient way to prepare a frustration-free state under the conditions of QLLL. A series of results showed that the answer is affirmative if all local Hamiltonians commute [40, 9, 37]. Recently, Gilyén and Sattath improved the previous constructive results by designing an algorithm that works efficiently under Shearer’s bound for non-commuting terms as well .
Therefore, the following three closely related problems beg answers:
Tight region for QLLL: complete characterization of the interior of QLLL, , for a given interaction bipartite graph . Here the interior is the set of vectors such that any local Hamiltonians with relative dimensions and interaction bipartite graph are frustration free. As Shearer’s bound has been shown to be a sufficient condition for QLLL , a fundamental question here is whether Shearer’s bound is tight. If it is tight indeed, there are several striking consequences. Firstly, the tightness implies Gilyén and Sattath’s algorithm  converges up to the tight region. Moreover, the geometrization theorem  says that given the interaction bipartite graph, dimensions of qudits, and dimensions of local Hamiltonians, either all such Hamiltonian are frustration free, or almost all such Hamiltonians are not. If Shearer’s bound is indeed tight for QLLL, by geometrization theorem we have that the quantum satisfiability for almost all Hamiltonians with large enough qudits can be completely characterized by the lattice gas partition function. Meanwhile, the lattice gas critical exponents can be directly applied to the counting of the ground state entropy of almost all quantum Hamiltonians in the frustration free regime. Thus, the tightness means a lot for transferring insights from classical statistical mechanics into the quantum complexity domain .
Tight region for CLLL: complete characterization of the interior of CLLL, , for a given interaction bipartite graph . Here the interior is the set of vectors such that any commuting Hamiltonians with relative dimensions and interaction bipartite graph are frustration free. It is immediate that the interior of QLLL is a subset of the interior of CLLL for any . An interesting question is whether the containment is proper. There are a series of results on the algorithms for preparing a frustration-free state for commuting Hamiltonians under the conditions of QLLL [40, 9, 37]. Thus if the containment turns out to be proper, it is possible to design algorithms for commuting Hamiltonians which is still efficient beyond the conditions of QLLL, e.g., Shearer’s bound. Tight region for CLLL begs characterization not only because the various applications in CLH, but also it may help us to understand the role of non-commutation plays in the quantum world.
Critical thresholds for LLLs: determining the critical probability threshold of VLLL and the critical relative dimension thresholds of CLLL and QLLL. Here the critical thresholds of LLLs are the minimum probability such that holds for some with probability vector and the minimum relative dimension such that some with relative dimension vector is not frustration free. Rather than other boundary probability vectors or relative dimension vectors, the symmetric boundary vector where all the elements are equal is much more often considered by physicists [45, 41, 3, 38] and computer scientists [46, 21, 31, 15, 14, 32, 20]. Sattath et al.  conjectured that the tight regions of VLLL and QLLL are different. If this conjecture turns out to be true, the next question is how large the gap is. A lower bound on the gap between VLLL and QLLL, especially that in the symmetric direction, constitutes a quantitative analysis of the relative power of quantum. Though we have the complete characterizations of LLLs, it still needs new ideas to quantify the critical thresholds and their gaps, because the mathematical characterizations, such as Shearer’s inequality system and the programme for VLLL, are usually hard to solve [21, 23].
1.1 Results and Discussion
In this paper, we mainly concentrate on the following three problems: the tight region for QLLL, the tight region for CLLL, and tight bounds for symmetric VLLL, CLLL and QLLL. We provide the complete answer for the first problem and partial answers for the other two problems. Our results show that Shearer’s bound, which is tight for abstract-LLL, is also tight for QLLL. The CLLL behaves very different from QLLL, i.e., the interior of CLLL goes beyond Shearer’s bound generally. Moreover, we also provide a lower bound on the critical thresholds of VLLL and CLLL, which are strictly larger than that of ALLL and QLLL on lattices. The main results are listed and discussed as follows.
In this work, the interaction bipartite graph of Hamiltonians and the classical event-variable graph are both denoted by bipartite graph . We call the vertices in the left vertices and those in the right vertices. Usually, we will indicate the left vertices with “” and the right vertices with “”. In , there may be two vertices with the same index , one is the left vertex and the other is the right vertex. In this paper, it will always have no ambiguity to identify which one it is from the context.
We prove several results on the regions or thresholds beyond Shearer’s bound. And “cycle” turns out to be an important structure. We say left vertices form a -cyclic graph where if and only if the induced subgraph containing exact these left vertices and their neighbors (i.e., the adjacent right vertices) can be transformed to a cycle by deleting the right vertices with degree 1. Furthermore, if all the neighbors of these left vertices are with degree at most 2, we call the -cyclic graph 2-discrete.
1.1.1 Tight region for QLLL
Shearer’s bound is tight for QLLL
Theorem 1.3 (Informal).
Given an interaction bipartite graph and relative projector ranks for all ,
If is below Shearer’s bound, then . For qudits with proper dimensions, this lower bound can be achieved by almost all Hamiltonians acting on these qudits, conforming with and satisfying for all . Moreover, for qudits with large enough dimensions, we have for almost all such Hamiltonians, where can be arbitrarily small as the dimensions of qudits go to infinity.
Otherwise, for qudits of proper dimensions, almost all Hamiltonians acting on these qudits, conforming with and satisfying for all are not frustration free. Furthermore, for qudits with large enough dimensions, we have for almost all such Hamiltonians, where can be arbitrarily small as the dimensions of qudits go to infinity.
In contrast to the VLLL which goes beyond Shearer’s bound generally, QLLL is another example exhibiting the difference between the classical world and the quantum world. As mentioned above, Theorem 1.3 means that the position of the first negative fugacity zero of the lattice gas partition function is exactly the critical threshold of quantum satisfiability for almost all Hamiltonians with large enough qudits, and the relative dimension of the smallest satisfying subspace is exactly characterized by the independent set polynomial. Additionally, the above theorem also shows the tightness of Gilyén and Sattath’s algorithm , which prepares a frustration free state under Shearer’s bound.
Independently, Siddhardh Morampudi and Chris Laumann showed that Shearer’s bound is tight for a large class of graphs . Our result shows that Shearer’s bound is tight for any graph.
1.1.2 Tight region for CLLL
We partially depict the tight region of CLLL. We show that Shearer’s bound is tight for CLLL on trees and explicitly provide the relative dimension bounds. On the other hand, we also show that the tight region of CLLL can go beyond Shearer’s bound if its base graph has an induced cycle of length at least 4. To obtain this result, we first prove that the tight regions of CLLL and VLLL are the same for a large family of interaction bipartite graphs (see Theorem 4.8) by Bravyi and Vyalyi’s Structure Lemma . Then we generalize the tools for VLLL to CLLL, including a sufficient and necessary condition for deciding whether Shearer’s bound is tight and the reduction rules. At last, we combine these tools with the conclusions for VLLL from  to finish the proof.
Equal to Shearer’s bound on trees.
The researches on the boundaries of LLLs on the interaction bipartite graph which is a tree have a long history, including 1-D chains , regular trees [43, 24, 8, 38], and treelike bipartite graphs . For LLLs on trees, our results include: 1, We prove that Shearer’s bound is tight for CLLL by the reduction rules (see Theorem 4.15); 2, We calculate the bound for CLLL explicitly even considering the dimensions of qudits (see Theorem 4.16); 3, We calculate the tight bound for LLLs explicitly ignoring the dimensions of qudits. The tight bound is as follows.
Given an interaction bipartite graph which is a tree, without loss of generality, we can assume that the root is the right vertex. Furthermore, it is lossless as well to assume the leaves of the tree are right vertices, because adding right vertices as leaves do not change the boundary (see Theorem 4.13).
For any interaction bipartite graph which is a tree, we have . Given , if and only if there exists where if is a leaf of and for other .
We should mention that the above theorem is not an immediate corollary by Shearer’s bound. Shearer’s bound is difficult to solve in general. However, the explicit bound in the above theorem can be calculated efficiently. Combining with Theorem 1.4 which calculates the critical threshold of ALLL on -regular tree , we have the following corollary.
For the interaction bipartite graph which is a -regular tree, i.e., any left vertex is of degree and any right vertex is of degree , we have .
Beyond Shearer’s bound if the interaction bipartite graph contains cycles.
By coupling our tools for CLLL with the conclusions about VLLL , we can prove the following theorem.
For any interaction bipartite graph that some left vertices form a cycle, the tight region of CLLL goes beyond Shearer’s bound.
Our theorem implies that it is possible to design more specialized algorithm for CLLL which is still efficient beyond Shearer’s bound. Meanwhile, recall that Shearer’s bound is tight for QLLL, the above theorem shows that CLLL behaves very different from QLLL.
By Theorems 1.4 and 1.6, we can prove the following corollary, which gives an almost complete characterization of whether Shearer’s bound is tight for CLLL except when the base graph has only 3-cliques.
Given an interaction bipartite graph, Shearer’s bound is tight for CLLL if its base graph is a tree, and is not tight if its base graph has an induced cycle of length at least 4.
1.1.3 Critical thresholds for different LLLs
To determine the critical thresholds of a given interaction bipartite graph is of course a fundamental problem, and has been extensively studied [35, 43, 24, 8, 38, 12, 13, 4, 45]. Given an interaction bipartite graph , let be the critical thresholds for ALLL, VLLL, CLLL and QLLL, respectively. For simplicity, we may omit when it is clear for the context. Here, we investigate these four kinds of critical thresholds, particularly their relationships.
Lower bound for the gaps between critical thresholds.
As it has been proved that the tight bounds of VLLL and CLLL can go beyond Shearer’s bound, i.e., there are gaps between the tight bounds of VLLL and CLLL and Shearer’s bound. The next question is how large these gaps are. Our following theorem provides lower bounds for these gaps. Our contribution here is a general approach to study gaps quantitative. Though we only investigate the gaps between critical thresholds here, i.e., the gaps between the tight bounds of LLLs in the direction of symmetric probability vector, the techniques we provided in the proofs can be applied to other directions as well.
Given a dependency graph , let be the maximum degree of vertices in . Given an interaction bipartite graph , let be the distance between and in for any . Let be . Then, we have the following theorem.
Given a bipartite graph and a constant , if for any , there is another on a 2-discrete -cyclic graph where , then , and .
Theorem 1.8 provides the first lower bound on the gap between the critical threshold of VLLL and Shearer’s bound, which constitutes a quantitative study on the difference between the classical world and the quantum world. It shows that for any finite graph (i.e., ) containing a 2-discrete cyclic subgraph, and are exactly larger than and . By Theorem 1.8, we can obtain the following corollary for cycles, which has received considerable attention in the LLL literature [27, 23].
For any -cyclic graph, we have , and .
Critical thresholds separation on lattices.
Given a dependency graph , it naturally defines an interaction bipartite graph as follows. Regard each edge of as a variable (or a qudit) and each vertex as an event (or a local Hamiltonian). An event (or local Hamiltonian ) depends on a variable (or a qudit ) if and only if the vertex corresponding to (or ) is an endpoint of the edge corresponding to (or ). We consider the critical thresholds of for a given dependency graph . Many of such graphs studied in the literature [35, 12, 38, 13, 4, 24, 45, 8] can be embedded into an Euclidean space naturally, and usually have a translational unit in the sense that can be viewed as the union of periodic translations of . For example, a cycle of length 4 is a translational unit of the square lattice. The following is a direct application of Theorem 1.8.
Let be a graph embedded in an Euclidean space. If is a tree, then . Otherwise, has a translational unit containing a cycle, and we have , and where is the number of vertices of .
As a concrete example, since we have already known  for square lattice, then by Theorem 1.10, we have . Moreover, by exploiting the specific structure of the square lattice, we can obtain a refined bound: . We calculate the lower bounds on () for several of the most common lattices, as summarized in Table 1, which can then be used to obtain better lower bound on () exceeding directly.
|Lattice||lower bound on|
|1-D chain||1/4 ||0|
|Triangular||[12, 4, 45]|
|Square||0.1193 [13, 45]|
|Simple Cubic||0.0744 |
Organization The organization of this paper is as follows. Section 2 provides the definitions and notations. In Section 3, we prove that Shearer’s bound is tight for QLLL. Section 4 shows that the tight region of CLLL is generally beyond Shearer’s bound. In Section 5, we investigate the critical thresholds of different LLLs and provide lower bounds for the gaps between them.
2 Definitions and Notations
Let be a given interaction bipartite graph and be a given relative dimension vector. We will use boldface type, e.g., , for vectors. For any and of the same dimensions, we say if holds for any . We say if and holds for some . In this paper, we are only interested in finite dimensional quantum systems.
For the sake of simplicity and without loss of generality, we assume the relative dimensions are strictly positive, i.e., . Furthermore, in the whole paper except Section 3, we assume that is connected (hence so is the corresponding dependency graph). In Section 3, we argue for general instead, as disconnected may be involved in the inductive steps.
Definition 2.1 (Hilbert space of the qudits).
Let be the number of qudits. Then, the Hilbert space of the quantum system is an
th-order tensor productover . For any , let denote the Hilbert space of the qudits in . For example, . For any , let be the dimension of and be .
We say the qudit is classical, or a classical variable, if any Hamiltonians acting on it can be written as where is the computational basis of .
Definition 2.2 (Projectors, subspaces and relative dimensions).
Given a subspace , let be the projector onto . The relative dimension of to is defined as . For simplicity, we will omit “to ” and use if there is no ambiguity. It is easy to see that is a rational number. We say a set of subspaces is frustration free if do not span , and we will use to represent the vector . In this paper, the two terms ”subspaces” and ”projectors” will be used interchangeably.
is called a classical Hamiltonian if is diagonal with respect to the computational basis.
Definition 2.3 (Events and variables).
Let event set be a set of events fully determined by a set of mutually independent random variables . Then for each , there is a unique minimal subset that determines . We denote this set of variables by . Let be . For each , let ( for short when it is clear from the context) stand for the universal set of the possible values of .
Definition 2.4 (Neighbors).
Given a bipartite graph , let (or if is implicit) denote the neighbors of vertex in if which side this vertex belongs to is clear from the context. We say two left vertices are neighboring or adjacent if . We say a left vertex and a right vertex are neighboring or adjacent if .
Given a dependency graph and any , let and . We say two vertices are neighboring or adjacent if .
Definition 2.5 (Hamiltonians and events on graphs).
Given a bipartite graph , we say a set of local Hamiltonians conforms with , denoted by , if for any , acts trivially on qudits . Thus, we can write as where . Similarly, we can also define a set of events conforms with , denoted by .
Here, we usually call the interaction bipartite graph.
Definition 2.6 (Dependency graph of ).
Given a bipartite graph , the corresponding dependency graph of is defined as , where if and only if the left vertices are neighbors in .
Given a bipartite graph , we define the multivariate independence polynomial (or if is implicit) of as that of , i.e.,
Definition 2.7 (Maximum Degree).
Given a dependency graph , let be the maximum degree of vertices in . Given an interaction bipartite graph , let be .
Definition 2.8 (Induced subgraph).
Given a bipartite graph and any , let be the induced subgraph of on the left vertices in and the right vertices in . Given a dependency graph and any , let be the induced subgraph of on .
Definition 2.9 (Contained graph).
We say an interaction bipartite graph contains another , if there is some such that can be obtained from by deleting the right vertices with degree no more than one and relabeling the left vertices and the right vertices, respectively.
Intuitively, if and is contained in , then is the interaction bipartite graph for a subset of events in .
Definition 2.10 (Interior and Boundary).
The classical abstract interior of the dependency graph , , is the set of vectors such that for any event set with dependency graph and probability vector . It turns out to be exactly the set of probability vectors below Shearer’s bound. The abstract boundary of a graph , denoted by , is the set . Any is called an abstract boundary vector of .
Similarly, we can also define for a given bipartite graph . The definitions of would be a little different from the classical case, because the relative dimensions of Hamiltonians cannot be irrational (refer to Definition 4.1 for the details).
For simplify, we let and for a given bipartite graph .
Definition 2.11 (Critical Threshold).
Given a bipartite graph , the critical threshold for ALLL is the probability such that is in .
Similarly, we can also define critical thresholds for VLLL (denoted by ), QLLL (denoted by ) and CLLL (denoted by ). It is easy to see that .
Definition 2.12 (Random subspaces).
Given an interaction bipartite graph , an rational vector and an integer vector , we say a subspace set of is an instance of the setting if , and . We further say a subspace set of is a random instance of the setting if and for any , where is a random subspace of according to the Haar measure with .
3 QLLL: Shearer’s Bound is Tight
This section aims at proving Theorem 1.3. We first present several useful tools.
3.1 Tools for QLLL
The geometrization theorem is a useful tool established by Laumann et al. . With this theorem, we can show “almost all” just by showing the “existence”.
Theorem 3.1 (The geometrization theorem, adapted from ).
Given the interaction bipartite graph , dimension vector and relative dimensions , if there exist of the setting satisfying , then for random instance of the setting , we have with probability 1.
Another tool for depicting the tight region of QLLL is the multivariate independence polynomial. Recall that the for any interaction bipartite graph and , is defined as where is the set of all independent sets of by Definitions 1.1 and 2.6. For any , the independence polynomial of , , is defined similarly recalling that is an induced subgraph of .
By the definition of independence polynomial, it is easy to verify the following properties.
Given an interaction bipartite graph and , we have
for any , ,
if for the right vertex , then .
As shown in Theorem 1.2, the independence polynomial provides an exact characterization of , i.e., the tight criterion under which abstract version LLL holds. Another interesting property of the independence polynomial is as follows.
Lemma 3.3 (Lemma 31 in ).
Given an interaction bipartite graph and , there is a unique such that is on the boundary of Shearer’s bound, i.e.,
and for any ,
, for any .
Then we have the following corollary.
Given an interaction bipartite graph and , if , then there must be some such that and for any .
Proof. By Lemma 3.3, we have there is a unique such that , for any , and for any . Thus, we have because . Then, the corollary is immediate by letting .
The following properties of the independence polynomial will also be used.
Given an interaction bipartite graph and , if and for any , then
Proof. 12. For any , we have . In other words, is non-increasing as grows. Thus, we have for any , . Additionally, for any .
3. Suppose there exists such that there is no edge between and in , then . Hence we have either or , a contradiction.
3.2 Shearer’s Bound is Tight for QLLL
The following theorem shows that Shearer’s bound is tight for QLLL. As Shearer’s bound has been shown to be a lower bound on the relative dimension of the satisfying subspace , it remains to show this lower bound can be achieved.
Given an interaction bipartite graph and rational ,
if is above Shearer’s bound (i.e., s.t. ) , then there is some such that for random subspace of the setting , we have .
Otherwise, is below Shearer’s bound. Then there is some such that for random subspace of the setting , we have .
Recall that with the geometrization theorem, we can show “almost all” just by showing the “existence”. The proof of existence is by an inductive randomized construction. The following example is a good illustration of our main idea.
Let be a 4-cyclic graph, i.e., where . Let . Note that the base graph of is a cycle of length 4, hence . Our construction here is randomized: Let and
where is a random subspace of with ,
where is a random subspace of with ,
is a random subspace of with ,
is a random subspace of with .
Consider the subspace and the associated Hamiltonians . Note that the base graph of this subsystem is a 3-path and the independence polynomial becomes , then by the induction hypothesis, we have w.p.1; Similarly, it also holds that w.p.1. Therefore, span the whole space w.p.1. Now we have shown the existence of such a subspace. By applying geometrization theorem, we can prove Theorem 3.6 for the 4-cyclic graph and .
Now we are ready to prove Theorem 3.6.
Part (a). Note that if , by Corollary 3.4 there must be some such that and for any . Thus, without loss of generality we can assume that and for any . The proof is by induction on the number of left vertices in .
Basic: If the number of left vertices in is no more than 1, the theorem holds trivially.
Induction: We assume this theorem holds for any interaction bipartite graph where the number of left vertices is no more than . In the following, we prove that the theorem also holds for any graph where the number of left vertices is .
Let be of the minimal size such that . If , then the theorem is immediate by the induction hypothesis.
In the following we assume , thus for any