Algebras, Synchronous Games and Chromatic Numbers of Graphs
William Helton, Kyle P. Meyer, Vern I. Paulsen, Matthew Satriano
aa r X i v : . [ m a t h . OA ] M a r ALGEBRAS, SYNCHRONOUS GAMES ANDCHROMATIC NUMBERS OF GRAPHS
J. WILLIAM HELTON, KYLE P. MEYER, VERN I. PAULSEN,AND MATTHEW SATRIANO
Abstract.
We associate to each synchronous game an algebrawhose representations determine if the game has a perfect deter-ministic strategy, perfect quantum strategy or one of several otherstrategies. When applied to the graph coloring game, this leadsto characterizations in terms of properties of an algebra of variousquantum chromatic numbers that have been studied in the litera-ture. This allows us to develop a correspondence between variouschromatic numbers of a graph and ideals in this algebra which canthen be approached via Gr¨obner basis methods. Introduction
Given a graph G and a natural number c , there is a game called the c-coloring game of G and it is known that there is a perfect determin-istic strategy for this game if and only if G has a c-coloring. Thus, the chromatic number of G , χ ( G ) can be characterized as the least inte-ger for which a perfect deterministic strategy exists for the c-coloringgame. This led researchers to consider various kinds of probabilis-tic strategies for games, especially cases where the probabilities arosefrom the random outcomes of quantum experiments that were in finitedimensional entangled states, called quantum strategies . The least c forwhich the c-coloring game has such a perfect strategy became knownas the quantum chromatic number of a graph, denoted χ q ( G ). For anintroduction to this literature, see [5], [13]. There are some open ques-tions about the proper model for the set of probabilities that arise fromentangled quantum experiments and this has led to the study of sev-eral, potentially different, definitions of quantum chromatic numbersdenoted χ qa ( G ) and χ qc ( G ), see [22] and [21]. The first and second-named authors have been supported in part by NSF DMS-1500835 .The third and fourth-named authors are supported in part by NSERC DiscoveryGrants.
However, while there are many graphs G for which χ q ( G ) = χ ( G ),it is not yet known whether or not χ q , χ qa , and χ qc can assume differ-ent values. Furthermore, there are currently no general algorithms forcomputing these three quantum chromatic numbers.There is also an alternative characterization of χ ( G ) in terms ofgraph homomorphisms. If K c denotes the complete graph on c vertices,then χ ( G ) is the least integer c for which a graph homomorphism existsfrom G to K c . Again there is a corresponding graph homomorphismgame and several, potentially different definitions of quantum graphhomomorphisms defined in terms of the existence of perfect quantumstrategies for the graph homomorphism game. In each case, the least cfor which there exists a perfect quantum strategy for the graph homo-morphism game from G to K c is the corresponding quantum chromaticnumber. For more on this literature see, [13], [14], and [25].Given graphs G and H , C. Ortiz and the third author [19] affiliateda *-algebra to this pair, denoted A ( G, H ), and characterized the exis-tence of graph homomorphisms from G to H in terms of representationsof this algebra: [19] proves that A ( G, H ) has a non-zero homomorphisminto the complex numbers if and only if there exists a classical graphhomomorphism from G to H . Because the problem of determining if χ ( G ) ≤ A ( G, K ) has a non-trivialhomomorphism into the complex numbers is an NP-complete problem.Similarly, [19] also shows that there exists a quantum graph homo-morphism , as defined by [13] and [25], from G to H if and only if A ( G, H ) has a non-zero *-homomorphism into the matrices, that is,a non-zero finite dimensional representation. A result of [8] showsthat the problem of determining if χ q ( G ) ≤ A ( G, K ) has any non-zerofinite dimensional representations is NP-hard.Currently, the computational complexity of these other variations ofthe quantum chromatic number is unknown. The results of [19] char-acterize the existence of these other types of quantum graph homomor-phisms in terms of the existence of various types of traces on the alge-bra A ( G, H ). Thus, questions about the existence or non-existence ofthese various types of graph homomorphisms, and consequently quan-tum colorings, is reduced to questions about these algebras.1.1.
New types of chromatic numbers.
This leads us to a moredetailed study of these algebras, which is the main topic of this paper.Since the existence of these various types of representations can becharacterized in terms of whether or not certain ideals in these algebras are proper, we are naturally led to study new chromatic numbers ofa graph determined by the least integer c so that the type of idealin which we are interested is proper. Thus, whenever a certain typeof ideal is proper, we can use that as the definition of a new type ofgraph homomorphism and obtain a new algebraic chromatic number ofa graph, which we then try to relate to prior chromatic numbers. Threeof these new parameters are denoted χ alg ( G ), χ hered ( G ), and χ lc ( G ).One goal of this paper is to study the properties of these new chromaticnumbers.In general, no algorithms are known for computing the quantumchromatic numbers of a graph. One advantage to χ alg ( G ) is that itscomputation reduces to a Gr¨obner basis problem in a non-commutativealgebra. However, using a machine-aided proof, we are able to showthat χ alg ( G ) ≤ χ hered fromthe other quantum chromatic numbers, while computing this param-eter can be done approximately with a combination of algebra andoptimization (semidefinite programming).Since all the earlier notions of quantum graph homomorphism andthe corresponding quantum chromatic numbers, were defined in termsof the existence of perfect quantum strategies for certain games, webegin by identifying a certain family of games for which we can carryout the construction of [19] and affiliate an algebra to the game suchthat the existence of various types of representations of the algebradetermines whether or not various types of perfect quantum strategiesexist for the game. For this family of games it is possible that all ofthe notions of perfect quantum strategies coincide and are equivalentto a certain hereditary ideal in this algebra being proper.1.2. Outline of paper.
In Section 2, we provide some background ongames and strategies, introduce the family of synchronous games whichare the games for which we can extend the results of [19] and constructan algebra.In Section 3, we introduce the *-algebra of a synchronous game,the chromatic numbers χ alg ( G ) and χ hered ( G ) and prove some of theirproperties.In Section 4, we focus on the case of 1, 2, and 3 colors.In Section 5, we show that removing the assumption that the alge-bra be a free *-algebra instead of just a free algebra, changes nothingessential. We also show that it is enough to study these algebras over Q instead of C . This allows us to use a Gr¨obner basis approach. J. W. HELTON, K. P. MEYER, V. I. PAULSEN, AND M. SATRIANO
In Section 6, we present the details of the machine-assisted proofthat χ alg ( G ) ≤ locally commuting chromatic number χ lc ( G ).2. Synchronous Games and Strategies
We lay out some definitions and a few basic properties of games andstrategies. We will primarily be concerned with the c -coloring gameand the graph homomorphism game.2.1. Definitions of games and strategies.
By a two-person finiteinput-output game we mean a tuple G = ( I A , I B , O A , O B , λ ) where I A , I B , O A , O B are finite sets and λ : I A × I B × O A × O B → { , } is a function that represents the rules of the game, sometimes calledthe predicate. The sets I A and I B represent the inputs that Aliceand Bob can receive, and the sets O A and O B , represent the outputsthat Alice and Bob can produce, respectively. A referee selects a pair( v, w ) ∈ I A × I B , gives Alice v and Bob w , and they then produceoutputs (answers), a ∈ O A and b ∈ O B , respectively. They win thegame if λ ( v, w, a, b ) = 1 and lose otherwise. Alice and Bob are allowedto know the sets and the function λ and cooperate before the game toproduce a strategy for providing outputs, but while producing outputs,Alice and Bob only know their own inputs and are not allowed to knowthe other person’s input. Each time that they are given an input andproduce an output is referred to as a round of the game.We call such a game synchronous provided that: (i) Alice and Bobhave the same input sets and the same output sets, which we denoteby I and O , respectively, and (ii) λ satisfies: ∀ v ∈ I, λ ( v, v, a, b ) = ( a = b a = b , that is, whenever Alice and Bob receive the same inputs then they mustproduce the same outputs. To simplify notation we write a synchronousgame as G = ( I, O, λ ).A graph G is specified by a vertex set V ( G ) and an edge set E ( G ) ⊆ V ( G ) × V ( G ), satisfying ( v, v ) / ∈ E ( G ) and ( v, w ) ∈ E ( G ) = ⇒ ( w, v ) ∈ E ( G ). The c-coloring game for G has inputs I A = I B = V ( G ) and outputs O A = O B = { , ..., c } where the outputs are thoughtof as different colors. They win provided that whenever Alice and Bob receive adjacent vertices, i.e., ( v, w ) ∈ E , their outputs are dif-ferent colors and when they receive the same vertex they must outputthe same color. Thus, ( v, w ) ∈ E ( G ) = ⇒ λ ( v, w, a, a ) = 0 , ∀ a , λ ( v, v, a, b ) = 0 , ∀ v ∈ V ( G ) , ∀ a = b and the rule function is equal to 1for all other tuples. It is easy to see that this is a synchronous game.Given two graphs G and H , a graph homomorphism from G to H is afunction f : V ( G ) → V ( H ) with the property that ( v, w ) ∈ E ( G ) = ⇒ ( f ( v ) , f ( w )) ∈ E ( H ). The graph homomorphism game from G to H has inputs I A = I B = V ( G ) and outputs O A = O B = V ( H ) . Theywin provided that whenever Alice and Bob receive inputs that are anedge in G , then their outputs are an edge in H and that whenever Aliceand Bob receive the same vertex in G they produce the same vertex in H . This is also a synchronous game.A deterministic strategy for a game is a pair of functions, h : I A → O A and k : I B → O B such that if Alice and Bob receive inputs( v, w ) then they produce outputs ( h ( v ) , k ( w )). A deterministic strategywins every round of the game if and only if ∀ ( v, w ) ∈ I A × I B , λ ( v, w, h ( v ) , k ( w )) = 1 . Such a strategy is called a perfect deterministic strategy .It is not hard to see that for a synchronous game, any perfect deter-ministic strategy must satisfy, h = k . In particular, a perfect determin-istic strategy for the c-coloring game for G is a function h : V ( G ) →{ , ..., c } such that ( v, w ) ∈ E ( G ) = ⇒ h ( v ) = h ( w ). Thus, a perfectdeterministic strategy is precisely a c-coloring of G . Similarly, a per-fect deterministic strategy for the graph homomorphism is precisely agraph homomorphism.Finally, it is not difficult to see that if K c denotes the complete graphon c vertices then a graph homomorphism exists from G to K c if andonly if G has a c-coloring. This is because any time ( v, w ) ∈ E ( G )then a graph homomorphism must send them to distinct vertices in K c . Indeed, the rule function for the c-coloring game is exactly thesame as the rule function for the graph homomorphism game from G to K c .A random strategy for such a game is a conditional probabilitydensity p ( a, b | v, w ), which represents the probability that, given inputs( v, w ) ∈ I A × I B , Alice and Bob produce outputs ( a, b ) ∈ O A × O B .Thus, p ( a, b | v, w ) ≥ v, w ) , X a ∈ O A ,b ∈ O B p ( a, b | v, w ) = 1 . J. W. HELTON, K. P. MEYER, V. I. PAULSEN, AND M. SATRIANO
In this paper we identify random strategies with their conditionalprobability density, so that a random strategy will simply be a condi-tional probability density p ( a, b | v, w ).A random strategy is called perfect if λ ( v, w, a, b ) = 0 = ⇒ p ( a, b | v, w ) = 0 , ∀ ( v, w, a, b ) ∈ I A × I B × O A × O B . Thus, for each round, a perfect strategy gives a winning output withprobability 1.We next discuss local random strategies, which are also sometimescalled classical , meaning not quantum. They are obtained as follows:Alice and Bob share a probability space (Ω , P ), for each input v ∈ I A ,Alice has a random variable, f v : Ω → O A and for each input w ∈ I B ,Bob has a random variable, g w : Ω → O B such that for each roundof the game, Alice and Bob will evaluate their random variables at apoint ω ∈ Ω via a formula that has been agreed upon in advance. Thisyields conditional probabilities, p ( a, b | v, w ) = P ( { ω ∈ Ω | f v ( ω ) = a, g w ( ω ) = b } ) . This will be a perfect strategy if and only if ∀ ( v, w ) , P ( { ω ∈ Ω | λ ( v, w, f v ( ω ) , g w ( ω )) = 0 } ) = 0 , or equivalently, ∀ ( v, w ) , P ( { ω ∈ Ω | λ ( v, w, f v ( ω ) , g w ( ω )) = 1 } ) = 1 . If we have a perfect local strategy and setΩ = ∩ v ∈ I A ,w ∈ I B { ω ∈ Ω | λ ( v, w, f v ( t ) , g w ( t )) = 1 } , then P (Ω ) = 1 since I A and I B are finite sets; in particular, Ω isnon-empty. If we choose any ω ∈ Ω and set h ( v ) = f v ( ω ) and k ( w ) = g w ( ω ), then it is easily checked that this is a perfect deterministicstrategy.Thus, a perfect classical random strategy exists if and only if a per-fect deterministic strategy exists. An advantage to using a perfectclassical random strategy over a perfect deterministic strategy, is thatit is difficult for an observer to find a deterministic strategy even afterobserving the outputs of many rounds.The idea behind nonlocal games is to allow a larger set of con-ditional probabilities, namely, those that can be obtained by allowingAlice and Bob to run quantum experiments to obtain their outputs.Definitions of these various sets of probability densities, includingloc, q, qa, qc, vect, nsb can be found in [19, Section 6] or [21], sowe will avoid repeating them here. We only remark that if for t ∈ { loc, q, qa, qc, vect, nsb } we use C t to denote the corresponding set ofconditional probabilities, then it is known that C loc ( C q ⊆ C qa ⊆ C qc ( C vect ( C nsb . The sets C q , C qa and C qc represent three potentially different mathe-matical models for the set of all probabilities that can arise as outcomesfrom entangled quantum experiments. The question of whether or not C qa = C qc for any number of experiments and any number of outputs isknown to be equivalent to Connes’ embedding conjecture due to resultsof [20].We say that p ( a, b | v, w ) is a perfect t-strategy for a game providedthat it is a perfect strategy that belongs to the corresponding set ofprobability densities.Given a graph G we set χ t ( G ) equal to the least c for which thereexists a perfect t-strategy for the c-coloring game for G . The aboveinclusions imply that χ ( G ) = χ loc ( G ) ≥ χ q ( G ) ≥ χ qa ( G ) ≥ χ qc ( G ) ≥ χ vect ( G ) ≥ χ nsb ( G ) . Currently, it is unknown if there are any graphs that separate χ q ( G ) , χ qa ( G )and χ qc ( G ) or whether these three parameters are always equal. Exam-ples of graphs are known for which χ ( G ) > χ q ( G ), for which χ qc ( G ) >χ vect ( G ) and for which χ vect ( G ) > χ nsb ( G ). For details, see [5], [22]and [21]. Other versions of quantum chromatic type graph parametersappear in [1] and a comparison of those parameters with χ qa and χ qc can be found in [1, Section 1].Similarly, we say that there is a t-homomorphism from G to H ifand only if there exists a perfect t-strategy for the graph homomor-phism game from G to H and it is unknown if q-homomorphisms,qa-homomorphisms and qc-homomorphisms are distinct or coincide.Finally, we close this section by showing that it is enough to considerso-called symmetric games. Note that in a synchronous game there isno requirement that λ ( v, w, a, b ) = 0 = ⇒ λ ( w, v, b, a ) = 0. That is,the rule function does not need to be symmetric in this sense. Thefollowing shows that it is enough to consider synchronous games withthis additional symmetry.Given G = ( I, O, λ ) a synchronous game, we define λ s : I × I × O × O → { , } by setting λ s ( v, w, a, b ) = λ ( v, w, a, b ) λ ( w, v, b, a ) and set G s = ( I, O, λ s ). Then it is easily seen that G s is a synchronous gamewith the property that λ s ( v, w, a, b ) = 0 ⇐⇒ λ s ( w, v, b, a ) = 0.2.2. A few properties of strategies.
In the remainder of this sec-tion, we prove the following slight extension of [22].
J. W. HELTON, K. P. MEYER, V. I. PAULSEN, AND M. SATRIANO
Proposition 2.1.
Let G = ( I, O, λ ) be a synchronous game and let p ( a, b | v, w ) = h h v,a , k w,b i be a perfect vect-strategy for G , where thevectors h v,a and k w,b are as in the definition of a vector correlation(see [19, 6.15] ). Then h v,a = k v,a , ∀ v ∈ I, a ∈ O .Proof. By definition, for each v ∈ I the vectors { h v,a : a ∈ O } aremutually orthogonal and { k v,a : a ∈ O } are mutually orthogonal. So,1 = X a,b ∈ O p ( a, b | v, v ) = X a ∈ O p ( a, a | v, v ) = X a ∈ O h h v,a , k v,a i ≤ X a ∈ O k h v,a kk k v,a k ≤ (cid:0) X a ∈ O k h v,a k (cid:1) / (cid:0) X a ∈ O k k v,a k (cid:1) / = 1 . Thus, the inequalities are equalities, which forces h v,a = k v,a for all v ∈ I and all a ∈ O . (cid:3) Corollary 2.2.
Let G = ( I, O, λ ) be a synchronous game and let t ∈{ loc, q, qa, qc, vect } . If p ( a, b | v, w ) is a perfect t-strategy for G , then p ( a, b | v, w ) = p ( b, a | w, v ) for all v, w ∈ I and all a, b ∈ O .Proof. If p ( a, b | v, w ) is a perfect t-strategy, then it is a perfect vect-strategy and hence there exist vectors as in the definition such that, p ( a, b | v, w ) = h h v,a , h w,b i = h h w,b , h v,a i = p ( b, a | w, v ) , where the middle equality follows since the inner products are assumedto be non-negative. (cid:3) This corollary readily yields the following result.
Proposition 2.3.
Let G = ( I, O, λ ) be a synchronous game and let t ∈ { loc, q, qa, qc, vect } . Then p ( a, b | v, w ) is a perfect t-strategy for G if and only if p ( a, b | v, w ) is a perfect t-strategy for G s . The *-algebra of a synchronous game
We begin by constructing a *-algebra, defined by generators and rela-tions, that is affiliated with a synchronous game. The existence or non-existence of various types of perfect quantum strategies for the gamethen corresponds to the existence or non-existence of various types ofrepresentations of this algebra. This leads us to examine various idealsin the algebra.3.1.
Relations generators and the basic *-algebra.
Let G = ( I, O, λ )be a synchronous game and assume that the cardinality of I is | I | = n while the cardinality of O is | O | = m . We will often identify I with { , ..., n − } and O with { , ..., m − } . We let F ( n, m ) denote the free product of n copies of the cyclic group of order m and let C [ F ( n, m )]denote the complex *-algebra of the group. We regard the group al-gebra as both a *-algebra, where for each group element g we have g ∗ = g − , and as an (incomplete) inner product space, with the groupelements forming an orthonormal set and the inner product is given by h f, h i = τ ( f h ∗ ) , where τ is the trace functional.For each v ∈ I we have a unitary generator u v ∈ C [ F ( n, m )] suchthat u mv = 1. If we set ω = e πi/m then the eigenvalues of each u v isthe set { ω a : 0 ≤ a ≤ m − } . The orthogonal projection onto theeigenspace corresponding to ω a is given by(3.1) e v,a = 1 m m − X k =0 (cid:0) ω − a u v (cid:1) k , and these satisfy 1 = m − X a =0 e v,a and u v = m − X a =0 ω a e v,a . The set { e v,a : v ∈ I, ≤ a ≤ m − } is another set of generators for C [ F ( n, m )].We let I ( G ) denote the 2-sided *-ideal in C [ F ( n, m )] generated bythe set { e v,a e w,b | λ ( v, w, a, b ) = 0 } and refer to it as the ideal of the game G . We define the *-algebraof G to be the quotient A ( G ) = C [ F ( n, m )] / I ( G ) . A familiar case occurs when we are given two graphs G and H and G is the graph homomorphism game from G to H . Then A ( G ) = A ( G, H ), where the algebra on the right hand side is the algebra in-troduced in [19], so we shall continue that notation in this instance.Recall that A ( G, K c ) is then the algebra of the c-coloring game for G . Definition 3.1.
We say that a game has a perfect algebraic strategyif A ( G ) is nontrivial. Given graphs G and H, we write G alg −→ H if A ( G, H ) is nontirvial. We define the algebraic chromatic number of Gto be χ alg ( G ) = min { c | A ( G, K c ) is nontrivial } The following is a slight generalization of [19, Theorem 4.7].
Theorem 3.2.
Let G = ( I, O, λ ) be a synchronous game. (1) G has a perfect deterministic strategy if and only if there existsa unital *-homomorphism from A ( G ) to C . (2) G has a perfect q-strategy if and only if there exists a unital*-homomorphism from A ( G ) to B ( H ) for some non-zero finitedimensional Hilbert space. (3) G has a perfect qc-strategy if and only if there exists a unitalC*-algebra C with a faithful trace and a unital *-homomorphism π : A ( G ) → C .Hence, if G has a perfect qc-strategy, then it has a perfect algebraicstrategy and so χ qc ( G ) ≥ χ alg ( G ) for every graph G .Proof. We start with the third statement. Since the game is synchro-nous, any perfect strategy p ( a, b | v, w ) must also be synchronous. By[21, Theorem 5.5], any synchronous density is of the following form: p ( a, b | v, w ) = τ ( E v,a E w,b ), where τ : C → C is a tracial state for a a uni-tal C*-algebra C generated by projections { E v,a } satisfying P a E v,a = I for all v .If we take the GNS representation [9] of C induced by τ , then theimage of C under this representation will be a quotient of C with all thesame properties and the additional property that τ is a faithful traceon the quotient.Now if, in addition, p ( a, b | v, w ) belongs to the smaller family of per-fect q-strategies, then by [21, Theorem 5.3] the C*-algebra C will befinite dimensional. Hence, the second statement follows.Finally, if p ( a, b | v, w ) belongs to the smaller family of perfect loc-strategies, then the C*-algebra C will be abelian, and hence, the firststatement follows. (cid:3) Remark 3.3.
It is also possible to characterize the existence of perfectqa-strategies, but the proof is a bit long for here: G has a perfect qa-strategy if and only if there exists a unital *-homomorphism of A ( G ) into the von Neumann algebra R ω . Putting an order on our *-algebra.
The *-algebra C [ F ( n, m )]also possesses an order defined as follows: let P be the cone generatedby all elements of the form f ∗ f for f ∈ C [ F ( n, m )] . If h, k ∈ C [ F ( n, m )]are self-adjoint elements, we write h ≤ k if k − h ∈ P . Next, notice that P induces a cone on A ( G ), which we regard as the positive elements,by setting A ( G ) + = { p + I ( G ) : p ∈ P} . Given two self-adjoint elements h, k ∈ A ( G ), we again write h ≤ k ifand only if k − h ∈ A ( G ) + . In the language of Ozawa [20] this makes A ( G ) into a semi-pre-C*-algebra . A self-adjoint vector subspace V ⊆ C [ F ( n, m )] is called hereditary provided that 0 ≤ f ≤ h and h ∈ V implies that f ∈ V . Problem 3.4.
Let G be a synchronous game. Find conditions on thegame so that the 2-sided ideal I ( G ) is hereditary. Later we will see an example of a game such that I ( G ) is not hered-itary. The following result shows why the hereditary condition is im-portant. Proposition 3.5.
Let G be a synchronous game and let I ( G ) be theideal of the game. Then I ( G ) is a hereditary subspace of C [ F ( n, m )] ifand only if (cid:0) A ( G ) + (cid:1) ∩ (cid:0) − A ( G ) + (cid:1) = (0) .Proof. Let x = x ∗ ∈ C [ F ( n, m )]. We begin by characterizing when theequivalence class x + I ( G ) is contained in A ( G ) + ∩ (cid:0) − A ( G ) + (cid:1) . Bydefinition, this occurs if and only if there are elements p = p ∗ , q = q ∗ in I ( G ) such that x + p ≥ − x + q ≥
0. This is equivalent to0 ≤ x + p ≤ p + q .Now suppose that x = x ∗ and that the equivalence class x + I ( G ) isnon-zero in A ( G ). If the class is contained in (cid:0) A ( G ) + (cid:1) ∩ (cid:0) − A ( G ) + (cid:1) then choosing p and q as in the previous paragraph, the element x + p demonstrates that I ( G ) is not hereditary.Conversely, if I ( G ) is not hereditary, then there exists x = x ∗ / ∈ I ( G )and q ∈ I ( G ) such that 0 ≤ x ≤ q . The inequality 0 ≤ x implies that x + I ( G ) ∈ A ( G ) + , while 0 ≤ q − x implies that q − x + I ( G ) = − x + I ( G ) ∈ A ( G ) + . Clearly, this element is non-zero. (cid:3) Given a subspace V we let V h denote the smallest hereditary sub-space that contains V and call this space the hereditary closure of V . We define the hereditary *-algebra of the game G to be thequotient A h ( G ) = C [ F ( n, m )] / I h ( G ) . Note that A h ( G ) is a quotient of A ( G ). Definition 3.6.
We say that a game has a perfect hereditary strategyif A h ( G ) is nontrivial. Given graphs G and H, we write G hered −→ H if A h ( G, H ) is nontrivial. We define the hereditary chromatic number ofG by χ hered ( G ) = min { c | A h ( G, K c ) is nontrivial } . We define the positive cone in A h ( G ) by setting A h ( G ) + = { p + I h ( G ) : p ∈ P} , so that A h ( G ) is also a semi-pre- C ∗ -algebra. The following is immedi-ate: Proposition 3.7.
Let G be a synchronous game, then (cid:0) A h ( G ) + (cid:1) ∩ (cid:0) − A h ( G ) + (cid:1) = (0) . Thus, the “positive” cone on A h ( G ) is now a proper cone and A h ( G )is an ordered vector space. Proposition 3.8. If K n hered −→ K c then n ≤ c . Consequently, χ hered ( K n ) = n. Proof.
Suppose, that K n hered −→ K c . We let E v,i = e v,i + I h ( K n , K c ) ∈A h ( K n , K c ), where A h is defined as in §
3. Then we have that P c − i =0 E v,i = I for all v. Set P i = P v E v,i . Since E v,i E w,i = 0 , we have that P i = P ∗ i = P i . Hence, Q i = I − P i is also a “projection” in the sense that Q i = Q ∗ i = Q i . Now P i P i = cI − P i Q i . But also, X i P i = X v X i E v,i = nI. Hence, P i Q i = P i Q i = ( c − n ) I. By definition, P i Q i ∈ A h ( G, H ) + .If c < n , then ( c − n ) I ∈ − ( A h ( G, H ) + ) and by Proposition 3.7, wehave I = 0; that is, 1 ∈ I h ( K n , K c ) which contradicts our hypothesisthat K n hered −→ K c . Hence, c ≥ n . This shows that χ hered ( K n ) ≥ n .For the other inequality, note that if K n hered −→ K c , then χ hered ( K n ) ≤ c . By the results of [19], if G is any graph with c = χ qc ( G ), thenthere is a unital *-homomorphism from A ( G, K c ) into a C*-algebrawith a trace. The kernel of this homomorphism is a hereditary idealand so must contain I h ( G, K c ). Hence, this latter ideal is proper and c ≥ χ hered ( G ). Thus, χ hered ( K n ) ≤ χ qc ( K n ) ≤ χ ( K n ) = n . Hence, n = χ hered ( K n ) ≤ c . (cid:3) The above proof also shows that:
Proposition 3.9. If K n alg → K c and − I / ∈ A ( K n , K c ) + , then n ≤ c . Pre- C ∗ -algebras. The next natural question is whether or not A h ( G ) is a pre- C ∗ -algebra in the sense of [20]. The answer is thatthis cone will need to satisfy one more hypothesis. Definition 3.10.
Let G be a synchronous game and let I c ( G ) denotethe intersection of the kernels of all unital ∗ -homomorphisms from C [ F ( n, m )] into the bounded operators on a Hilbert space (possibly 0dimensional) that vanish on I ( G ) . Let A c ( G ) = C [ F ( n, m )] / I c ( G ) . Proposition 3.11.
Let G be a synchronous game. Then I h ( G ) ⊆ I c ( G ) and I c ( G ) = { x ∈ C [ F ( n, m )] : x ∗ x + I ( G ) ≤ ǫ I ( G ) , ∀ ǫ > , ǫ ∈ R } . There exists a (non-zero) Hilbert space H and a unital *-homomorphism π : C [ F ( n, m )] → B ( H ) that vanishes on I ( G ) if and only if A c ( G ) =(0) .Proof. The kernel of every *-homomorphism is a hereditary ideal andthe intersection of hereditary ideals is a hereditary ideal, hence I c ( G )is a hereditary ideal containing I ( G ). So, I h ( G ) ⊆ I c ( G ).We have that x ∈ I c ( G ) if and only if x + I ( G ) is in the kernelof every *-homomorphism of A ( G ) into the bounded operators on aHilbert space. In [20, Theorem 1] it is shown that this is equivalent to x + I ( G ) being in the “ideal of infinitesimal elements” of A ( G ), whichis the ideal defined by the right-hand side of the above formula. Thelast result comes from the fact that the ideal of infinitesimal elementsis exactly the intersection of the kernels of all such representations. (cid:3) Definition 3.12.
We say that a game G has a perfect C*-strategy provided that A c ( G ) is nontrivial. Following [19] , we write G C ∗ −→ H provided that for given graphs G and H , the algebra A c ( G, H ) is non-trivial. We define the C*-chromatic number of G to be χ C ∗ ( G ) = min { c | A c ( G, K c ) is nontrivial } . The following is immediate.
Proposition 3.13.
Let G be a graph. Then χ qc ( G ) ≥ χ C ∗ ( G ) ≥ χ hered ( G ) ≥ χ alg ( G ) . This motivates the following question.
Problem 3.14.
Let G be a synchronous game. Is I c ( G ) = I h ( G ) ? Problem 3.15. If I h ( G ) = C [ F ( n, m )] , then does there exist a non-zero Hilbert space H and a unital *-homomorphism, π : C [ F ( n, m )] → B ( H ) such that π ( I h ( G )) = (0) , that is, if I h ( G ) = C [ F ( n, m )] , then is I c ( G ) = C [ F ( n, m )] ? Determining if an ideal is hereditary.
Here we mention someliterature on determining if an ideal I is hereditary and the issue ofcomputing its “hereditary closure.” In the real algebraic geometry lit-erature, a hereditary ideal is called a real ideal. For a finitely generatedleft ideal I in R ( F ( k )) the papers [2, 3] present a theory and a numer-ical algorithm to test (up to numerical error) if I is hereditary. Thealgorithm also computes the “hereditary radical” of I . The computeralgorithm relies on numerical optimization (semidefinite programming)and hence it is not exact but approximate.For two sided ideals [4] and [10] contain some theory. Also the firstauthor and Klep developed and crudely implemented a hereditary ideal algorithm under NCAlgebra. However, it is too memory consuming tobe effective, so we leave this topic for future work.A moral one can draw from this literature is that computing hered-itary closures is not broadly effective at this moment.3.5. Clique Numbers.
The clique number of a graph ω ( G ) is definedas the size of the largest complete subgraph of G. It is not hard to seethat G contains a complete subgraph of size c if and only if there is agraph homomorphism from K c to G. Hence, there is a parallel theoryof quantum clique numbers that we shall not pursue here, other thanto remark that for each of the cases t ∈ { loc, q, qa, qc, C ∗ , hered, alg } we define the t-clique number of g by ω t ( G ) = max { c | K c t −→ G } , so that ω ( G ) = ω loc ( G ) ≤ ω q ( G ) ≤ ω qa ( G ) ≤ ω qc ( G ) ≤ ω C ∗ ( G ) ≤ ω hered ( G ) ≤ ω alg ( G ) . Lovasz [12] introduced his theta function ϑ ( G ) of a graph. Thefamous Lovasz sandwich theorem [7] says that for every graph G , if G denotes its graph complement, then ω ( G ) ≤ ϑ ( G ) ≤ χ ( G ). In [19,Proposition 4.2] they showed the following improvement of the Lovaszsandwich theorem: ω C ∗ ( G ) ≤ ϑ ( G ) ≤ χ C ∗ ( G ) . We shall show later, χ alg ( K ) = 4, while ϑ ( K ) = 5 . Hence the sand-wich inequality fails for the algebraic version.This motivates the following problem:
Problem 3.16. Is ω hered ( G ) ≤ ϑ ( G ) ≤ χ hered ( G ) for all graphs? The case of 1, 2 and 3 colors
It is a classic result that deciding if χ ( G ) ≤ χ q ( G ) ≤ χ C ∗ ( G ) ≤ , χ hered ( G ) ≤ , and χ alg ( G ) ≤
3. Addressing the first two inequalities would require oneto compute I c ( G, K ) and I h ( G, K ), and unfortunately these idealscontain elements not just determined by simple algebraic relations.However, studying A ( G, K ) and A ( G, K ) is rewarding, as we shallsee now see. Throughout the section, we use the notation E v,i = e v,i + I ( K n , K c ). Proposition 4.1.
Let G be a graph. Then χ alg ( G ) = 1 if and only if G is the empty graph. Hence, χ alg ( G ) = 1 ⇐⇒ χ ( G ) = 1 . Proof.
For each vertex we only have one idempotent E v, and sincethese sum to the identity, necessarily E v, = I . But if there is an edge( v, w ) then I = I · I = E v, E w, = 0. (cid:3) Proposition 4.2.
Let G be a connected graph on more than one vertex.Then χ alg ( G ) = 2 ⇐⇒ χ ( G ) = 2 . Proof.
First assume that χ alg ( G ) = 2. Fix a vertex v and set P = E v, and P = E v, . Note P + P = I. Let ( v, w ) ∈ E ( G ), then P E w, = 0and P E w, = 0 . Hence, E w, = ( P + P ) E w, = P E w, and similarly, E w, = E w, P . Also, P = P ( E w, + E w, ) = P E w, = E w, . Thus, whenever ( v, w ) ∈ E ( G ) , then E v,i = E w,i +1 , i.e., there are twoprojections and they flip. Since G is connected, by using a path from v to an arbitrary w we see that { E w, , E w, } = { P , P } . Now we wish to 2-color G . Define the color of any vertex w to be 0if E w, = P and 1 if E w, = P . This yields a 2-coloring, and since G is connected on more than one vertex, there is no 1-coloring, showing χ ( G ) = 2.Conversely, if χ ( G ) = 2 then G is not the empty graph. Since1 ≤ χ alg ( G ) ≤
2, by the previous result, χ alg ( G ) = 2. (cid:3) Proposition 4.3. If ( v, w ) ∈ E ( G ) then E v,i E w,j = E w,j E v,i ∈ A ( G, K ) for all i, j . In particular, if G is complete, then A ( G, K ) is abelian.Proof. For 0 = E v, E v, = E v, ( E w, + E w, + E w, ) E v, = E v, E w, E v, . Similarly, E v,i E w,j E v,k = 0 whenever { i, j, k } = { , , } . Now E w, = ( E v, + E v, + E v, ) E w, ( E v, + E v, + E v, ) = E v, E w, E v, + E v, E w, E v, . Similarly, E w,j = E v,j +1 E w,j E v,j +1 + E v,j +2 E w,j E v,j +2 . Hence, for i = j , E v,i E w,j = E v,i E w,j E v,i = E w,j E v,i , while when i = j , E v,i E w,i = 0 = E w,i E v,i . (cid:3) Theorem 4.4. χ alg ( K j ) = j for j = 2 , , . Proof.
We have that χ alg ( K ) = 2 , by Proposition 4.2. Now if χ alg ( K ) =2 then by Proposition 4.2, we see χ ( K ) = 2 , which is a contradiction.Hence, 3 ≤ χ alg ( K ) ≤ χ ( K ) = 3 . Finally, if χ alg ( K ) = 3, then by Proposition 4.3, we have that A ( K , K ) is a non-zero abelian complex *-algebra. But every uni-tal, abelian ring contains a proper maximal ideal M , and forming thequotient we obtain a field F . The map λ → λ M embeds C asa subfield. Now we use the fact that A ( K , K ) is generated by pro-jections and that the image of each projection in F is either 0 or 1 inorder to see that the range of the quotient map is just C . Thus, F = C and we have a unital homomorphism π : A ( K , K ) → C and againusing the fact that the image of each projection is 0 or 1 and that theprojections commute, we see that π is a *-homomorphism. Hence, by[19, Theorem 4.12], we have χ ( K ) ≤
3, a contradiction.Thus, 3 < χ alg ( K ) ≤ χ ( K ) = 4 and the result follows. (cid:3) Corollary 4.5. I ∈ I ( K , K ) . *-Algebra versus Free Algebra The original motivation for the construction of the algebra of a gamecomes from projective quantum measurement systems which are alwaysgiven by orthogonal projections on a Hilbert space, i.e., operators sat-isfying E = E = E ∗ . This is why we have defined the algebra of agame to be a *-algebra. But a natural question is whether or not onereally needs a *-algebra or is there simply a free algebra with relationsthat suffices? In this section we show that as long as one introduces thecorrect relations then the assumption that the algebra be a *-algebrais not necessary.To this end let F ( nm ) := C h x v,a | ≤ v ≤ n − , ≤ a ≤ m − i bethe free unital complex algebra on nm generators and let B ( n, m ) = F ( nm ) / I ( n, m ) where I ( n, m ) is the two-sided ideal generated by x v,a − x v,a , ∀ v, a ; 1 − m − X a =0 x v,a , ∀ v ; x v,a x v,b , ∀ v, ∀ a = b. We let p v,a denote the coset of x v,a in the quotient so that p v,a = p v,a , ∀ v, a ; 1 = m − X a =0 p v,a , ∀ v ; p v,a p v,b = 0 , ∀ v, ∀ a = b. Proposition 5.1.
There is an isomorphism π : B ( n, m ) → C [ F ( n, m )] with π ( p v,a ) = e v,a , ∀ v, a , where e v,a are defined as in the previous sec-tion.Proof. Let ρ : F ( nm ) → C [ F ( n, m )] be the unital algebra homomor-phism with ρ ( x v,a ) = e v,a Then ρ vanishes on I ( n, m ) and so induces aquotient homomorphism π : B ( n, m ) → C [ F ( n, m )]. It remains to showthat π is one-to-one.To this end set ω = e πi/m and let y v = P m − a =0 ω a p v,a . It is readilychecked that y v y mv = P m − a =0 ω − am p v,a = 1. Since p v,a = m P m − k =0 ( ω − a y v ) k we have that { y v : 0 ≤ v ≤ n − } generates B ( n, m ).Now by the universal property of C [ F ( n, m )] there is a homomor-phism γ : C [ F ( n, m )] → B ( n, m ) with γ ( u v ) = y v and hence, this is theinverse of π . (cid:3) If a unital algebra contains 3 idempotents, p , p , p with p + p + p = 1, then necessarily p i p j = 0 for i = j . To see this note that p + p = 1 − p is idempotent. Squaring yields that p p + p p = 0.Thus, 0 = p ( p p + p p ) = p p + p p p and 0 = ( p p + p p ) p = p p p + p p , from which it follows that p p = p p and so 2 p p = 0and the claim follows.Hence, when m = 3 the condition that x v,a x v,b = 0 , ∀ a = b is aconsequence of the other hypotheses and is not needed in the definitionof the ideal I ( n, m ).However, Heydar Radjavi [23] has constructed a set of 4 idempotentsin a unital complex algebra which sum to the identity but for which p i p j = 0 for i = j . Thus, for m ≥
4, it is necessary to include therelation x v,a x v,b in the ideal in order to guarantee that the quotient B ( n, m ) is isomorphic to C [ F ( n, m )], since these products are 0 in thelatter algebra.If a set of self-adjoint projections on a Hilbert space, P , ..., P m sum tothe identity then it is easily checked that they project onto orthogonalsubspaces and so P i P j = 0 , ∀ i = j . Thus, in any C*-algebra whenself-adjoint idempotents sum to the identity, their pairwise productsare 0. But the situation is not so clear for self-adjoint idempotents ina *-algebra and we have not been able to resolve this question. So weask: Problem 5.2.
Let A be a unital *-algebra and let p , ..., p m satisfy p i = p i = p ∗ i and p + · · · + p m = 1 . Then does it follow that p i p j = 0 , ∀ i = j ? Corollary 5.3.
Let G = ( I, O, λ ) be a symmetric synchronous gamewith | I | = n and | O | = m . Then A ( G ) is isomorphic to the quotient of F ( nm ) by the 2-sided ideal generated by x v,a − x v,a , ∀ v, a ; 1 − m − X a =0 x v,a , ∀ v and x v,a x w,b , ∀ v, w, a, b such that λ ( v, w, a, b ) = 0 . Note that x v,a x v,b for a = b is in the ideal since λ ( v, v, a, b ) = 0. Wenote that the hypothesis that the game be symmetric is needed, sincein a *-algebra, the condition that x v,a x w,b = 0 implies that x w,b x v,a = 0,while this relation would not necessarily be met in the quotient of thefree algebra. Corollary 5.4.
A symmetric synchronous game G = ( I, O, λ ) has aperfect algebraic strategy if and only if 1 is not in the 2-sided ideal of F ( nm ) generated by x v,a − x v,a , ∀ v, a ; 1 − m − X a =0 x v,a , ∀ v, and x v,a x w,b , ∀ v, w, a, b such that λ ( v, w, a, b ) = 0 . Change of Field.
The following is important for Gr¨obner basiscalculations. Let 1 ∈ K ⊆ C be any subfield. We set F K ( nm ) equal tothe free K -algebra on nm generators x v,i , so that F C ( nm ) = F ( nm ).Given any symmetric synchronous game G with n inputs and m out-puts, we let I K ( G ) ⊆ F K ( nm ) be the 2-sided ideal generated by x v,a − x v,a , ∀ v, a ; 1 − m − X a =0 x v,a , ∀ v, and x v,a x w,b , ∀ v, w, a, b such that λ ( v, w, a, b ) = 0 . We let A K ( G ) = F K ( nm ) / I K ( G ). By Corollary 5.4, G has a perfectalgebraic strategy if and only if 1
6∈ I C ( G ), or equivalently A C ( G ) = 0.We show that this computation is independent of the field K . Proposition 5.5. If K is a field containing Q , then A K ( G ) = A Q ( G ) ⊗ Q K . Furthermore, A K ( G ) = 0 if and only if A Q ( G ) = 0 .Proof. By definition, we have a short exact sequence0 → I Q → F Q → A Q → Q -vector spaces. Since K is flat over Q , we obtain a short exactsequence 0 → I Q ⊗ Q K → F Q ⊗ Q K → A Q ⊗ Q K → . Since the generators are independent of the field, one checks that F Q ⊗ Q K = F K and that the image of I Q ⊗ Q K → F K is equal to I K . Sincethis latter map is injective, we see I Q ⊗ Q K = I K . Hence, the aboveshort exact sequence shows A K ( G ) = A Q ( G ) ⊗ Q K .Lastly, K / Q is faithfully flat. So, A K ( G ) = 0 if and only if A Q ( G ) =0. (cid:3) The case of 4 colors
This section gives a machine-assisted proof which analyzes 4 alge-braic colors. We prove:
Theorem 6.1.
For any graph G , we have χ alg ( G ) ≤ . This theorem is equivalent to the statement that for any graph G , theideal satisfies 1
6∈ I ( G, K ). We will prove this statement through theuse of (noncommutative) Gr¨obner bases. For a brief effective expositionto noncommutative Gr¨obner basis algorithms, see Chapter 12.3 [6] or[16, 24, 11].For those readers already familiar with (commutative) Gr¨obner bases,we explain the key differences with the noncommutative setting. Let I = ( p , . . . p k ) be a two-sided ideal, and prescribe a monomial order.A noncommutative Gr¨obner basis B of I is a set of generators suchthat the leading term of any element of I is in the monomial idealgenerated the leading terms of B . A noncommutative Gr¨obner basis isproduced in the same way as in the commutative case. Let m j be theleading term of p j and notice that any two m j , m k have as many as4 possible least common multiples, each of which produces syzygyies.One repeatedly produces syzygyies and reduces to obtain a Gr¨obnerbasis in the same way as the commutative setting. However, unlike thecommutative case, a Gr¨obner basis can be infinite. Very fortunatelythe Gr¨obner bases that arise in our coloring computations below arefinite. The key property we use is that p is in I if and only if thereduction of p by a Gr¨obner basis for I yields 0.Recall that I ( G, K ) is generated by the following relations: x v,i x v,j ∀ i = j ; 1 − X i =0 x v,i ∀ v ; x v,i x w,i ∀ ( v, w ) ∈ E ( G ) , ∀ i. To prove Theorem 6.1 we will make use of the following theorem:
Theorem 6.2.
For any n ≥ a Gr¨obner basis for I ( K n , K ) underthe graded lexographic ordering with (6.1) x , < x , < x , < x , < x , < x , < . . . < x n − , consists of relations of the following forms: (1) x v,i x v,j with i, j ≤ i = j (2) x v,i − x v,i with i ≤ x v, + x v, + x v, + x v, − x v,i x w,i with v = w , i ≤ x v, x w, + x v, x w, + x v, x w, + x v, x w, + x v, x w, + x v, x w, − x v, − x v, − x v, − x w, − x w, − x w, + 1 with v = w (6) x v, x w, x v, − x v, x w, x v, − x v, x w, x v, − x v, x w, x v, − x v, x w, x v, − x v, x w, x v, + x v, x w, + x v, x w, + x v, x w, + x v, x w, + x w, x v, + x w, x v, + x w, x v, + x w, x v, − x v, − x v, − x v, − x w, − x w, − x w, + 1 with v = w (7) x v, x w, x x, − x v, x w, x x, − x v, x w, x x, − x v, x w, x x, − x v, x w, x x, − x v, x w, x x, + x v, x x, + x v, x w, + x v, x w, +2 x v, x x, +2 x v, x x, + x v, x w, + x v, x w, +2 x v, x x, + x v, x x, + x w, x x, + x w, x x, + x w, x x, + x w, x x, − x v, − x v, − x v, − x w, − x w, − x w, − x x, − x x, − x x, + 2 with v = w = x = v Specifically I ( K n , K ) has a Gr¨obner basis that does not contain 1and thus
6∈ I ( K n , K ) . Remark 6.3.
Each of the relations (1) – (7) correspond to a set of re-lations obtained by taking all choices of v, w, x in V ( K n ) . Howeverbecause of the monomial ordering chosen, the leading terms are alwaysof the forms: (1) x v,i x v,j (2) x v,i (3) x v, (4) x v,i x w,j (5) x v, x w, (6) x v, x w, x v, (7) x v, x w, x x, Additionally for every ≤ i ≤ , all the vertices of K n which appearin the terms of relation (i) also appear in the leading term of (i). Proof.
Let the ideal generated by these relations be denoted by J , wewill first show that these relations form a Gr¨obner basis for J , andthen show that J = I ( K n , K ).Before we begin our calculations pertaining to an algebra over C wenote that all of the coefficients that appear will be in Q . Section 5.1bears on this.To see that these relations form a Gr¨obner basis we must show thatthe syzygy between any two polynomials in this list is zero when re-duced by the list. First by Remark 6.3 each of the relations has vari-ables corresponding to at most three different vertices of K n and reduc-ing by a relation will not introduce variables corresponding to differentvertices. Thus when calculating and reducing the syzygy between anytwo relations, variables corresponding to at most 6 vertices of K n willbe involved. Therefore we can verify that all syzygies reduce to zeroby looking at the case n = 6 which we verify using NCAlgebra 5.0 andNCGB running under Mathematica (see notebook QCGB-9-20-16.nb,available at: https://github.com/NCAlgebra/UserNotebooks ). Thisproves that the relations (1) – (7) form a Gr¨obner basis.We now show that J = I ( K n , K ). We will first show that all of thegenerators of J are contained in I ( K n , K ). The elements of types (1),(3), and (4) are self-evidently in I ( K n , K ) since they are elements ofthe generating set of I ( K n , K ). For type (2) we note that under therelations generating I ( K n , K ) that x v,i (1 − X j =0 x v,j ) = x v,i − x v,i − X j = i x v,i x v,j = x v,i − x v,i , and thus elements of type (2) are in I ( K n , K ). For type (5) we usethe relations generating I ( K n , K ) to get that x v, x w, = (1 − x v, − x v, − x v, )(1 − x w, − x w, − x w, )= x v, x w, + x v, x w, + x v, x w, + x v, x w, + x v, x w, + x v, x w, − x v, − x v, − x v, − x w, − x w, − x w, + 1 . Finally type (6) is obtained by reducing( x v, x w, + x v, x w, + x v, x w, + x v, x w, + x v, x w, + x v, x w, − x v, − x v, − x v, − x w, − x w, − x w, + 1) x v, − x v, ( x w, x v, )using the relations of types (1)–(5), and type (7) is obtained by reducing( x v, x w, + x v, x w, + x v, x w, + x v, x w, + x v, x w, + x v, x w, − x v, − x v, − x v, − x w, − x w, − x w, + 1) x x, − x v, ( x w, x x, ) using the relations of types (1)–(5). These two reductions are verifiedwith Mathematica in QCGB-9-20-16.nb. Thus all of the generatingrelations of J are in I ( K n , K ) and we have that J ⊂ I ( K n , K ).Next we will show that all the generators of I ( K n , K ) are containedin J . The only generating relations of I that are not immediately seento be in J are x v, x v,j , x v,i x v, , and x v, x w, . To see that x v,i x v, is in J we consider x v,i ( x v, + x v, + x v, + x v, − J since ( x v, + x v, + x v, + x v, −
1) is in J , andwhen multiplied out all terms except x v,i x v, are in J , and thus x v,i x v, is in J , similarly x v, x v,j is in J . Finally, we consider the equation x v, x w, = ( x v, + x v, + x v, + x v, − x w, + x w, + x w, + x w, − − ( x v, x w, + x v, x w, + x v, x w, + x v, x w, + x v, x w, + x v, x w, − x v, − x v, − x v, − x w, − x w, − x w, + 1) − x v, x w, − x v, x w, − x v, x w, , the right-hand side is a sum of relations in J and is thus in J , and thusthe left-hand side is also in J , specifically x v, x w, is in J . Therefore allof the generating relations of I ( K n , K ) are in J , so that I ( K n , K ) ⊂J . Since we have shown inclusion both ways, we have that I ( K n , K ) = J and we are done. (cid:3) Lemma 6.4. If G , H are graphs such that V ( H ) = V ( G ) and E ( H ) ⊃ E ( G ) , then I ( H, K m ) ⊃ I ( G, K m ) and thus
6∈ I ( H, K m ) = ⇒ ( G, K m ) .Proof. The relations generating I ( H, K m ) contains the relations gen-erating I ( G, K m ) and thus the result follows. (cid:3) Proof of Theorem 6.1.
Let G be a graph on n vertices. By Theorem6.2, 1
6∈ I ( K n , K ). Additionally E ( G ) ⊂ E ( K n ), and thus by Lemma6.4, 1
6∈ I ( G, K ). Therefore χ alg ( G ) ≤ (cid:3) Problem 6.5.
We do not know the complexity of deciding if χ alg ( G ) =4 , i.e., of deciding if ∈ I ( G, K ) . The Locally Commuting Algebra
Our analysis of “algebraic colorability” shows χ alg is too coarse aparameter to provide much information about graphs, after all everygraph will be algebraically colorable by at most 4 colors, and so has χ alg has very little in common with any of the quantum chromatic numbers.This problem does not occur with the hereditary chromatic number, which may be equal to the usual quantum chromatic number for allgraphs, but unfortunately determining the elements of the hereditaryclosure of an ideal is difficult. In the forthcoming section, we addfurther physically motivated algebraic relations to I ( G, H ), in order toobtain a chromatic number that is more amenable to algebraic analysiswhile still retaining a quantum flavor.The new relations are additional commutation relations in A ( G, H )and yield a new algebra A lc ( G, H ) that we call the locally commutingalgebra . This yields a new type of chromatic number, χ lc . One goal ofthis section is to prove χ lc ( K n ) = n .Since our algebras were initially motivated by quantum chromaticnumbers, it is natural to look to quantum mechanics for further rela-tions to impose. In the case of a graph, we can imagine each vertex ascorresponding to a laboratory and think of two vertices as connectedwhenever those laboratories can conduct a joint experiment. In thiscase, all of the measurement operators for the two labs should commute,i.e., whenever ( v, w ) is an edge, then the commutator [ e v,i , e w,j ] := e v,i e w,j − e w,j e v,i = 0. Note that this commutation rule is exactly therule that we were able to derive in the case of three colors in Proposi-tion 4.3. This motivates the following definitions. Definition 7.1.
Let G = ( I, O, λ ) be a synchronous game with | I | = n and | O | = m . We say that v, w ∈ I are adjacent and write v ∼ w provided that v = w and there exists a, b ∈ O such that λ ( v, w, a, b ) = 0 .We define the locally commuting ideal of the game to be the 2-sidedideal I lc ( G ) in C [ F ( n, m )] generated by the set { e v,a e w,b | λ ( v, w, a, b ) = 0 } ∪ { [ e v,a , e w,b ] | v ∼ w, ∀ a, b ∈ O } . We set A lc ( G ) = C [ F ( n, m )] / I lc ( G ) and call this the locally commut-ing algebra of G .In the case that G and H are graphs and G is the graph homomor-phism game from G to H we set I lc ( G, H ) = I lc ( G ) and A lc ( G, H ) = A lc ( G ) . We write G lc → H provided that I lc ( G, H ) = C [ F ( n, m )] and set χ lc ( G ) = min { c | G lc → K c } . We similarly define ω lc ( G ) = max { c | K c lc → G } . Note that in the case of the graph homomorphism game from G to H we have that I = V ( G ) and v ∼ w ⇐⇒ ( v, w ) ∈ E ( G ). Thus,the relationship ∼ extends the concept of adjacency to the inputs of ageneral synchronous game.Thus, A lc ( G, K c ) is the universal *-algebra generated by self-adjointprojections { E v,i : v ∈ V ( G ) , ≤ i ≤ c } satisfying • P ci =1 E v,i = I, ∀ v, • v ∼ w = ⇒ E v,i E w,i = 0 , ∀ i, • v ∼ w = ⇒ [ E v,i , E w,j ] = 0 , ∀ i, j and χ lc ( G ) is the least c for which such a non-trivial *-algebra exists.We begin by showing that every graph homomorphism, in the usualsense, yields an lc -morphism: Lemma 7.2. If G → H , then G lc → H .Proof. Let φ : G → H be a graph homomorphism. We must show A lc ( G, H ) = 0. Consider the map A lc ( G, H ) → C sending e v,φ ( v ) to1 and e v,x to 0 for x = φ ( v ). It is easy to see this is a well-defined C -algebra map and hence surjective. As a result, A lc ( G, H ) = 0. (cid:3) Corollary 7.3.
We have χ lc ( G ) ≤ χ ( G ) and ω ( G ) ≤ ω lc ( G ) .Proof. There is a graph homomorphism G → K χ ( G ) so by Lemma 7.2,we have G lc → K χ ( G ) and hence χ lc ( G ) ≤ χ ( G ). The inequality for ω isshown in an analogous fashion. (cid:3) We are now ready to prove the main result of this section.
Theorem 7.4.
For every n ≥ , we have that χ lc ( K n ) = n .Proof. The case n = 1 is trivial to check. Assume that n ≥
2. Let uslabel the vertices by numbers 1 to n and set c = χ lc ( K n ) . Then c ≤ n and we want to show that c < n is impossible.Suppose that c < n , then since I = P ci =1 E k,i for 1 ≤ k ≤ n we havethat I = n Y k =1 (cid:0) c X i =1 E k,i (cid:1) = X i ,...,i n E ,i E ,i · · · E n,i n , where the sum is over all n -tuples with i j ∈ { , ..., c } . By the locallycommuting hypothesis, all of the above projections commute, so wemay re-order the sum in any fashion. Since c < n , by the pigeon-holeprinciple, each n -tuple must contain j, l with i j = i l = h . But then E j,h E l,h = 0. Hence, each product occurring in the above sum is 0and so the sum is 0. Thus, we have that I = 0, which shows that A lc ( K n , K c ) = (0), that is, I lc ( K n , K c ) = C [ F ( n, c )] for c < n . (cid:3) Problem 7.5.
We do not know if the Lov¨asz sandwich result holds inthis context, i.e., if ω lc ( G ) ≤ ϑ ( G ) ≤ χ lc ( G ) . Some basic properties of A lc and χ lc In this section we analyze the algebra A lc ( G, H ) more closely andobtain the value of χ lc ( G ) for a few select graphs. In particular, weare able to show, indirectly, that χ lc = χ q . Throughout this section weshall write ≃ to indicate that two algebras are isomorphic. We shalluse C n to denote the abelian algebra of complex-valued functions on n points.It is easy to check that A lc ( G, H ) is the quotient of C h e vx | v ∈ G, x ∈ H i by the ideal generated by the following relations:(1) P x ∈ H e vx = 1,(2) e vx = e vx ,(3) e vx e vy = 0 for x = y ,(4) e vx e wy = 0 if v ∼ w and x y , and(5) [ e vx , e wy ] = 0 for v ∼ w .In Lemma 7.2 we showed that graph homomorphisms induce lc -morphisms.We next show that we can “compose” lc -morphisms. Lemma 8.1. If G lc → H and H lc → K , then G lc → K .Proof. If A lc ( G, H ) and A lc ( H, K ) are non-zero, then we must provethat A lc ( G, K ) is non-zero as well. To see this, consider the map C h e vr | v ∈ G, r ∈ K i → A lc ( G, H ) ⊗ A lc ( H, K )given by e vr X x ∈ H e vx ⊗ e xr and suppose that it vanishes on I lc ( G, K ). Hence, there would be awell-defined map on the quotient, A lc ( G, K ) → A lc ( G, H ) ⊗ A lc ( H, K ) e vr X x ∈ H e vx ⊗ e xr . If 1 = 0 in A lc ( G, K ), then the same would be true in A lc ( G, H ) ⊗A lc ( H, K ), since this map sends units to units.Thus it remains to show that the above map vanishes on I lc ( G, K ).In order to do this, it is sufficient to check that each generating relationis sent to zero. This is easily checked, for example, X r ∈ K X x ∈ H e vx ⊗ e xr = X x ∈ H e vx ⊗ X r ∈ K e xr = X x ∈ H e vx ⊗ . Checking the other relations is left to the reader. (cid:3)
Corollary 8.2. If G lc → H , then χ lc ( G ) ≤ χ lc ( H ) .Proof. Let c = χ lc ( H ). Then we have H lc → K c and hence G lc → K c .Thus, χ lc ( G ) ≤ c = χ lc ( H ). (cid:3) We also have the following consequence of the proof of Lemma 8.1.
Theorem 8.3.
The assignment ( Graphs ) × ( Graphs ) −→ ( C -algebras )( G, H ) lc ( G, H ) is a functor, which is covariant in the first factor and contravariant inthe second; note that the category of graphs is with usual morphisms,not lc -morphisms.Proof. If φ : G → G ′ is a morphism, then we have a map A lc ( G, H ) →A lc ( G ′ , H ) given by e v,x e φ ( v ) ,x . On the other hand, if φ : H → K is a morphism, then we have H lc → K and so from the proof of Lemma8.1, we have A lc ( G, K ) → A lc ( G, H ) ⊗ A lc ( H, K ) . Since φ is a morphism of graphs, we have a map A lc ( H, K ) → C as in the proof of Lemma 7.2. Composing with the above, we have A lc ( G, K ) → A lc ( G, H ). Explicitly, this map is given by sending e vx ∈A lc ( G, K ) to P φ ( r )= x e vr . (cid:3) We now show how the functor A lc interacts with various naturalgraph operations. To begin, recall that if G is a graph, its suspensionΣ G is defined by adding a new vertex v and an edge from v to each ofthe vertices of G .Given an algebra A we shall let A c denote the algebra of c -tupleswith entries from A , i.e., the tensor product A ⊗ C c ≃ A ⊕ c where C c can be identfied with the algebra of C -valued functions on c points. Proposition 8.4.
Let G and H be any graphs, and let H ni be the non-isolated vertices. For y ∈ H ni we let N y denote the neighborhood of y , i.e., the induced subgraph of H with vertices adjacent to y ; notice y / ∈ N y unless y has a self-edge. Then A lc (Σ G, H ) ≃ M y ∈ H ni A lc ( G, N y ) . In particular, if H is vertex transitive and y is any vertex of H withneighborhood N , then A lc (Σ G, H ) ≃ A lc ( G, N ) | H | . Proof.
Let u be the new vertex added to Σ G , i.e., u ∈ Σ G \ G . Since u is adjacent to every vertex of G , we see e ux commutes with e vy for all v ∈ G and x, y ∈ H . Furthermore, the defining relations of A lc tell us e ux e uy = δ x,y e ux where δ denotes the Kronecker delta function. So, A lc (Σ G, H ) ≃ A lc ( G, H )[ e ux ] / ( X x e ux − , e ux e uy = δ x,y e ux ) . In other words, the e ux for x ∈ H are commuting orthogonal idempo-tents, which shows A lc (Σ G, H ) ≃ M y ∈ H A lc ( G, H ) e uy ≃ M y ∈ H A lc ( G, H ) / ( e vx : x y ) , where the last equality comes from the fact that e vx e uy = 0 for x y .Now note that e vx remains non-zero in the quotient A lc ( G, H ) / ( e vx : x y ) if and only if x ∼ y . Thus, A lc ( G, H ) / ( e vx : x y ) ≃ A lc ( G, N y ) , which establishes the first assertion of the proposition. The secondassertion easily follows from the first since all neighborhoods are iso-morphic. (cid:3) Corollary 8.5.
For all graphs G , we have A lc (Σ G, K ) = 0 . If c ≥ ,then A lc (Σ G, K c ) ≃ A lc ( G, K c − ) c . Proof.
This is an immediate consequence of Proposition 8.4 using that K c is vertex transitive. (cid:3) Corollary 8.6.
For all graphs G , we have χ lc (Σ G ) = χ lc ( G ) + 1 .Proof. By the above isomorphism, the least c +1 such that A lc (Σ G, K c +1 ) =(0) is equal to the least c such that A lc ( G, K c ) = (0). (cid:3) Remark 8.7. In [15] an example of a graph G is given for which χ q (Σ G ) = χ q ( G ) . Hence, either χ lc (Σ G ) = χ q (Σ G ) or χ lc ( G ) = χ q ( G ) . Corollary 8.8. If c ≥ n , then A lc ( K n , K c ) ≃ C c ( c − ... ( c − n +1) . If c < n , then A lc ( K n , K c ) = 0 . Proof.
One easily checks that A lc ( K , G ) ≃ C | G | for any graph G . Inparticular, our desired statement holds for n = 1. The proof thenfollows from induction on n by applying Corollary 8.5 and using that K n = Σ K n − . (cid:3) Remark 8.9.
In Theorem 7.4, we proved χ lc ( K n ) = n . Corollary 8.8gives another proof of this result which is more refined: the corollarytells us the specific structure of A lc ( K n , K c ) whereas the theorem merelytells us it is non-zero. Using Proposition 8.4, we can easily understand iterated suspensions.In particular, we can understand any lc -map out of K n : Corollary 8.10. If H is a graph, then A lc ( K , H ) = C | H | and for n > , we have A lc ( K n , H ) ≃ M S ⊆ H C | N S | ( n − where S is an ( n − -clique and N S = { z ∈ H | z ∼ x ∀ x ∈ S } . Inparticular, if H is vertex transitive on c vertices, then A lc ( K n , H ) ≃ C c ( c − ... ( c − n +2) | N S | where S is any ( n − -clique in H .Proof. We leave the n = 1 case to the reader. Iteratively applyingProposition 8.4, we see A lc ( K n , H ) ≃ M x ∈ H A lc ( K n − , N x ) ≃ · · · ≃ M ( x n − ,...,x ,x ) A lc ( K , N x n − . . . N x N x )where the index of the direct sum runs over all sequences ( x n − , . . . , x , x )with x i +1 ∈ N x i N x i − . . . N x .We show by induction that the x , . . . , x i form an i -clique and that N x i N x i − . . . N x = N { x ,...,x i } . For i = 1 this is just the definition. For i >
1, observe that by construction x i ∈ N x i − . . . N x = N { x ,...,x i − } and since x , . . . , x i − forms an ( i − x , . . . , x i formsan i -clique. Next, N x i N { x ,...,x i − } is the set of z ∈ N { x ,...,x i − } that areadjacent to x i , which is the definition of N { x ,...,x i } .This shows A lc ( K n , H ) ≃ M ( x n − ,...,x ,x ) A lc ( K , N { x ,...,x n − } ) , and by the n = 1 case, each summand is isomorphic to | N { x ,...,x n − } | copies of C . Now notice that N { x ,...,x n − } is independent of the orderof the sequence, and hence this term arises ( n − If H is vertex transitive and contains an ( n − (cid:0) cn − (cid:1) such choices of an ( n − (cid:18) cn − (cid:19) ( n − | N S | = c ( c − . . . ( c − n + 2) | N S | many copies of C . (cid:3) We next consider how A lc interacts with the categorical product × .Recall that if H and K are graphs then H × K is a graph with vertexset V ( H ) × V ( K ) and where ( v, x ) ∼ ( w, y ) if and only if v ∼ H w and x ∼ K y . Theorem 8.11.
We have a natural isomorphism A lc ( G, H × K ) ≃ A lc ( G, H ) ⊗ A lc ( G, K ) , where × is the categorical product in graphs. Remark 8.12.
As shown in Theorem 8.3, we can view A lc as a functor.Theorem 8.11 can be interpreted as saying that the second factor ofthe functor A lc ( − , − ) preserves products. Recall that since the secondfactor is contravariant, preserving products means that it takes productsin graphs to coproducts in C -algebras, namely tensor products.Proof. Since we have a map H × K → H , Theorem 8.3 tells us that wehave a map A lc ( G, H ) → A lc ( G, H × K ). Similarly for K , and hencea natural map A lc ( G, H ) ⊗ A lc ( G, K ) → A lc ( G, H × K ). Explicitly, itis given by e α,v ⊗ X x ∈ K e α, ( v,x ) , ⊗ e α,y X w ∈ H e α, ( w,y ) . We now construct an inverse map given by A lc ( G, H × K ) → A lc ( G, H ) ⊗ A lc ( G, K ) e α, ( v,x ) e α,v ⊗ e α,x . Provided this is well-defined, it is indeed an inverse since e α,v ⊗ e α,x X y,w e α, ( v,y ) e α, ( w,x ) = e α, ( v,x ) , where the last equality uses the fact that e α, ( v,y ) e α, ( w,x ) = 0 unless( v, y ) = ( w, x ). Also, composing the two maps in the opposite orderyields the identity since e α,v ⊗ X x e α, ( v,x ) X x e α,v ⊗ e α,x = e α,v ⊗ X x e α,x = e α,v ⊗ . The rest of the proof is devoted to showing that the map we con-structed above is well-defined. First note that( e α,v ⊗ e α,x ) = e α,v ⊗ e α,x = e α,v ⊗ e α,x . Next, X ( v,x ) ∈ H × K e α,v ⊗ e α,x = X v e α,v ⊗ X x e α,x = 1 ⊗ . If ( v, x ) = ( w, y ) then without loss of generality v = w . So, e α, ( v,x ) e α, ( w,y ) ( e α,v ⊗ e α,x )( e α,w ⊗ e α,y ) = 0 ⊗ e α,x e α,y = 0 . Next assume α ∼ β . We need to check that the image of e α, ( v,x ) e β, ( w,y ) is equal to that e β, ( w,y ) e α, ( v,x ) . e α, ( v,x ) e β, ( w,y ) ( e α,v ⊗ e α,x )( e β,w ⊗ e β,y ) = e α,v e β,w ⊗ e α,x e β,y and since α ∼ β , this is equal to e β,w e α,v ⊗ e β,y e α,x = ( e β,w ⊗ e β,y )( e α,v ⊗ e α,x )which is the image of e β, ( w,y ) e α, ( v,x ) . Finally, we must show that if α ∼ β and ( v, x ) ( w, y ) then e α, ( v,x ) e β, ( w,y ) maps to 0. Since ( v, x ) ( w, y ),without loss of generality v w . Then the image of e α, ( v,x ) e β, ( w,y ) is e α,v e β,w ⊗ e α,x e β,y = 0since e α,v e β,w = 0. (cid:3) Problem 8.13.
In light of Remark 8.12, we ask if the functor A lc ( G, − ) preserves all finite limits. Given Theorem 8.11, this is equivalent toasking if it preserves equalizers. Problem 8.14 (Yoneda-type question) . Does the functor A lc ( G, − ) determine G ? Recall that the exponential of graphs K H is defined as follows: itsvertex set consists of all functions f : V ( H ) → V ( K ) and there is anedge f ∼ K H g if for all v ∼ H w we have f ( v ) ∼ K g ( w ). Note that if f is a graph homomorphism, then it has a self-edge.Within the category of graphs, the product is left adjoint to expo-nentiation, that is Hom( G × H, K ) = Hom(
G, K H ) . One can ask if this adjunction remains true for graphs with lc -morphisms. Problem 8.15.
If one allows H to have self edges, then A lc ( G, H ) canbe defined using the same relations at the beginning of this section. Onecan then define a map A lc ( G × H, K ) → A lc ( G, K H ) given by e ( α,v ) ,x X f ( v )= x e α,f . Is this map an isomorphism? In other words, are products and expo-nentials adjoints?
In addition to the categorical product × , there are several otherkinds of products on graphs, which we now consider. Both G (cid:3) H and G ⊠ H have vertex set V ( G ) × V ( H ). In the former, ( v, x ) ∼ ( w, y )if and only if v = w and x ∼ y , or v ∼ w and x = y . In the latter,( v, x ) ∼ ( w, y ) if and only if v ∼ w and x ∼ y , or v = w and x ∼ y ,or v ∼ w and x = y . The products (cid:3) and ⊠ are referred to as theCartesian and strong products, respectively.For any pair of graphs we have χ ( G (cid:3) H ) = max { χ ( G ) , χ ( H ) } . Weshow that the same is true for χ lc . Theorem 8.16 ( χ lc of Cartesian product) . For any graphs G and H ,we have χ lc ( G (cid:3) H ) = max { χ lc ( G ) , χ lc ( H ) } . Proof.
We have at least | H | maps G → G (cid:3) H , so Lemma 7.2 andCorollary 8.2 show χ lc ( G ) ≤ χ lc ( G (cid:3) H ). Similarly for H and somax { χ lc ( G ) , χ lc ( H ) } ≤ χ lc ( G (cid:3) H ). To prove the result, it now sufficesto show we have a map A lc ( G (cid:3) H, K c ) → A lc ( G, K c ) ⊗ A lc ( H, K c ) . Indeed, if A lc ( G, K c ) and A lc ( H, K c ) are non-zero, then so is A lc ( G (cid:3) H, K c )since the above map would send 0 to 0 and 1 to 1, and if 0 = 1in A lc ( G (cid:3) H, K c ), then 0 = 1 in A lc ( G, K c ) ⊗ A lc ( H, K c ), which isnot the case. Taking c = max { χ lc ( G ) , χ lc ( H ) } , this would then show χ lc ( G (cid:3) H, K c ) ≥ c .We now construct the above map. We define it by: e ( x,y ) ,k X i ∈ Z /c e x,i ⊗ e y,k − i and show it is well-defined. First suppose that ( x, y ) ∼ ( x ′ , z ) and k ℓ . Then k = ℓ and without loss of generality x = x ′ and y ∼ z .Then e ( x,y ) ,k e ( x,z ) ,ℓ X i,j e x,i e x,j ⊗ e y,k − i e z,ℓ − j = 0since e x,i e x,j = 0 if i = j , and if i = j , then k − i = ℓ − j and so e y,k − i e z,ℓ − j = 0. Next, if y ∼ z , then the images of e ( x,y ) ,k e ( x,z ) ,ℓ and e ( x,z ) ,ℓ e ( x,y ) ,k areequal since e ( x,y ) ,k e ( x,z ) ,ℓ X i,j e x,i e x,j ⊗ e y,k − i e z,ℓ − j and e y,k − i e z,ℓ − j = e z,ℓ − j e y,k − i as y ∼ z , and e x,i e x,j = δ ij e x,i = e x,j e x,i .We next see that X k e ( x,y ) ,k X i X k e x,i ⊗ e y,k − i = X i X k e x,i ⊗ e y,k = X i e x,i ⊗ X k e y,k = 1 ⊗ . If k = ℓ , then e ( x,y ) ,k e ( x,y ) ,ℓ X i,j e x,i e x,j ⊗ e y,k − i e y,ℓ − j = 0since e x,i e x,j = 0 if i = j , and if i = j , then k − i = ℓ − j and so e y,k − i e y,ℓ − j = 0.Lastly, e x,y ) ,k X i,j e x,i e x,j ⊗ e y,k − i e y,k − j = X i e x,i ⊗ e y,k − i since e x,i e x,j = 0 if i = j . Thus, e x,y ) ,k and e ( x,y ) ,k have the same image.This completes the proof that the map is well-defined. (cid:3) Lemma 8.17.
Given G lc → K and H lc → K ′ , we have G · H lc → K · K ′ forany · ∈ {× , (cid:3) , ⊠ } .Proof. It suffices to construct a map A lc ( G · H, K · K ′ ) → A lc ( G, K ) ⊗ A lc ( H, K ′ ) . We define it by e ( x,y ) , ( k,k ′ ) e x,k ⊗ e y,k ′ . One readily checks that this map is well-defined. For example, in thecase of the Cartesian product (cid:3) we show that if ( x, y ) ∼ ( z, w ) and( k, k ′ ) ( ℓ, ℓ ′ ), then e ( x,y ) , ( k,k ′ ) e ( z,w ) , ( ℓ,ℓ ′ ) maps to 0. Without loss ofgenerality, we can assume that x = z and y ∼ w . Then the imageis e x,k e x,ℓ ⊗ e y,k ′ e w,ℓ ′ , which is automatically 0 if k = ℓ . So, we mayassume k = ℓ , in which case k ′ ℓ ′ since ( k, k ′ ) ( ℓ, ℓ ′ ). But then e y,k ′ e w,ℓ ′ = 0. (cid:3) Corollary 8.18.
We have χ lc ( G ⊠ H ) ≤ χ lc ( G ) χ lc ( H ) .Proof. This follows immediately from Lemma 8.17 after observing that K n ⊠ K m = K nm . (cid:3) In [5], the authors showed that 8 = χ ( C ⊠ K ) = χ q ( C ⊠ K ) >χ vect ( C ⊠ K ) = 7. Thus, separating χ q from χ vect . Later, [21] showedthat χ qc ( C ⊠ K ) = 8, separating the potentially smaller χ qc from χ vect .We show below that χ lc ( C ⊠ K ) = 8 as well.Before considering C ⊠ K we begin with a simpler example. Example 8.19 ( C ⊠ K ) . Let G = C ⊠ K . It is easy to see that ω ( G ) = 4 and χ ( G ) = 5 , so a priori χ lc could be 4 or 5. We show χ lc ( C ⊠ K ) = 5 . We need to show that A lc ( G, K ) = 0 . The graph G is made up of 2pentagons stacked on top of each other. Let one of the pentagons havevertices x, y, z, w, s labeled clockwise and let the other pentagon havevertices x ′ , y ′ , z ′ , w ′ , s ′ with x and x ′ having the same neighbors. Forease of notation, we denote e v,i by v i . Note that X σ ∈ S ( s σ (3) s ′ σ (4) + s ′ σ (3) s σ (4) ) x σ (1) x ′ σ (2) y σ (3) y ′ σ (4) ( z σ (1) z ′ σ (2) + z ′ σ (1) z σ (2) ) . Multiplying on both the left and right by w w ′ , we obtain w w ′ = w w ′ ( s s ′ + s ′ s )( x x ′ + x ′ x )( y y ′ + y ′ y )( z z ′ + z ′ z ) w w ′ = 0 . Similarly, we find w i w ′ j = 0 for all i, j . As a result, X i,j w i w ′ j = 0 and so A lc ( G, K ) = 0 . Example 8.20 ( C ⊠ K ) . Let G = C ⊠ K . We see ω = 6 and χ = 8 ,so a priori χ lc could be 6, 7, or 8. We show χ lc ( C ⊠ K ) = 8 = χ ( C ⊠ K ) . We must show A lc ( G, K ) = 0 . We follow the same notational con-ventions as in Example 8.19. Let x, y, z, w, s be the vertices of C labeled clockwise and denote the next two copies of C by x ′ , . . . , s ′ resp. x ′′ , . . . , s ′′ where x , x ′ , x ′′ have the same neighbors in G . We alsolet v i = e v,i .As in the previous example, w w ′ w ′′ = w w ′ w ′′ SXY Zw w ′ w ′′ , where S = P i,j,k s i s ′ j s ′′ k and analogously for X, Y, Z . We show thatevery term occurring in the sum on the righthand side of the aboveequation is 0. The indices i, j, k occurring in the sum S must alllie in { , , , } otherwise the term vanishes (since it is multiplied by w w ′ w ′′ ). Our goal is to show that all terms in the sum in the righthand side vanish, so we can fix a summand in S and assume i, j, k equal , , respectively. Then the indices in X must be 3 of { , , , } . We also seethat the indices in Z must be 3 of { , , , } . Fix a summands x a x ′ b x ′′ c and z p z ′ q z ′′ r of X and Z , respectively. Then { , , . . . , } \ { a, b, c, p, q, r } has size at most 2. Therefore, every summand t of Y satisfies XtZ = 0 .So, w w ′ w ′′ = 0 , and analogously we see w i w ′ j w ′′ k = 0 for all i, j, k . So, X i,j w i w ′ j w ′′ k = 0 showing that A lc ( G, K ) = 0 . As a result, χ lc ( G ) = 8 . We end by posing the following:
Problem 8.21.
Since the definition of χ lc is not obviously related torepresentations on Hilbert spaces, it is unclear how to relate it to χ t for t ∈ { loc, q, qa, qc, vect } . Where does χ lc fit within this hierarchy? Even more specifically,
Problem 8.22.
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Department of Mathematics, UCSD
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