ON SENSITIVITY IN BIPARTITE CAYLEY GRAPHS

A BSTRACT . Huang proved that every set of more than half the vertices of the d -dimensional hypercube Q d induces a subgraph of maximum degree at least √ d , which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three inﬁnite families of Cayley graphs of unbounded degree that contain induced subgraphs of maximum degree 1 on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which ﬁrst counterexamples were shown recently by Lehner and Verret. The ﬁrst family consists of dihedrants and contains a sporadic counterexample encountered earlier by Lehner and Verret. The second family are star graphs, these are edge-transitive Cayley graphs of the symmetric group. All members of the third family are d -regular containing an induced matching on a d 2 d − 1 -fraction of the vertices. This is largest possible and answers a question of Lehner and Verret. Second, we consider Huang’s lower bound for graphs with subcubes and show that the corresponding lower bound is tight for products of Coxeter groups of type A n , I 2 (2 k + 1) , and most exceptional cases. We believe that Coxeter groups are a suitable generalization of the hypercube with respect to Huang’s question. Finally, we show that induced subgraphs on more than half the vertices of Levi graphs of projective planes and of the Ramanujan graphs of Lubotzky, Phillips, and Sarnak have unbounded degree. This gives classes of Cayley graphs with properties similar to the ones provided by Huang’s results. However, in contrast to Coxeter groups these graphs have no subcubes.


INTRODUCTION
Recently, Huang [20] proved the Sensitivity Conjecture [25] by showing that an induced subgraph on more than half of the vertices of the d-dimensional hypercube Q d has maximum degree at least √ d. For a graph G = (V, E) denote by α(G) the size of a largest independent set in G, by ∆(G) its maximum degree, and for a K ⊆ V by G[K] the subgraph induced by K. Define the sensitivity σ(G) of G as the minimum value ∆(G[K]) among all the K ⊆ V on more than α(G) vertices. Since in a regular bipartite G one has α(G) = |V | 2 , Huang's result can be expressed as Huang asks what can be said about σ(G) if G is a "nice" graph with high symmetry. Further, since by a result of Chung, Füredi, Graham, and Seymour [10] the bound for Q d is tight, he wonders for which graphs a tight bound on the sensitivity follows from his method.
The present paper studies both these questions by considering (simple, undirected 1 , right) Cayley graphs of groups to be "nice" with high symmetry. That is, for a group Γ and a subset C ⊆ Γ define Cay(Γ, C) with {x, y} ∈ E if and only if x −1 y ∈ C. First positive results into this direction were obtained by Alon and Zheng [3] for elementary abelian 2-groups. Then recently, Potechin and Tsang [26] showed that for every d-regular Cayley graph G of an abelian group any set of vertices of more than half the vertices induces a subgraph with maximum degree at least d/2 -hence answering Huang's question in the bipartite case. Moreover, they conjectured this lower bound to hold for Cayley graphs of general groups. However shortly after, Lehner and Verret [21] found a bipartite cubic Cayley graph G of a dihedral group with σ(G) = 1 < 3/2 -thus, rejecting the above conjecture. Moreover, they construct an infinite family of bipartite Cayley graphs of 2-groups of unbounded degree, with σ(G) = 1 for every member G of the family. Thus, concerning Huang's questions, σ(G) cannot be bounded from below by a function of the degree for general Cayley graphs.
In the fist part of the present paper, we give three more insensitive families of Cayley graphs, i.e., they have unbounded degree but σ(G) = 1 for all their members G.
The first family are bipartite dihedrants, i.e., Cayley graphs of the dihedral group (Theorem 2.1). The smallest member of this family is the graph presented in [21,Section 3] as well as the smallest non-cyclic, bipartite Cayley graph with σ = 1 among all groups.
The second family are the star-graphs [2], i.e., Cayley graphs of S n with respect to all transpositions containing 1. These graphs, that were initially motivated as an "attractive alternative" to the hypercube (see [1]) form a family of bipartite edge-transitive Cayley graphs. The first non-trivial member is the Nauru graph G (12,5), see Figure 1 and [14] for a beautiful collection of models. Another feature that distinguishes this family from the previous one is that they are Cayley graphs with respect to a minimal set of generators of the group. We show that besides their very high symmetry star graphs have sensitivity 1 (Theorem 3.3).
The third family consists of d-regular Cayley graphs that have an induced subgraph of maximum degree 1 on a d 2d−1 -fraction of the vertices (Theorem 4.1). This is largest possible in a d-regular graph and settles a question posed in [21,Remark 2]. In particular, we find the smallest such graphs and construct bipartite tight Cayley graphs by using the Kronecker double cover (Corollary 4.3).
The second part of the paper concerns the question when σ can be bounded from below in a tight way. A first answer to this could be that many groups including dihedral groups admit Cayley graphs that are isomorphic to Cayley graphs of abelian groups, see [24]. Hence, in the bipartite case their sensitivity admits a lower bound in term of their degree by [26]. Also, in [21,Remark 4], the authors describe their groups as close to abelian, (dihedral groups have a cyclic group of index 2, while 2-groups are nilpotent). They ask for a natural family of Cayley graphs of non-abelian groups for which σ grows in terms of the degree.
To this end consider the following easy consequence of Huang's result. If a bipartite Cayley graph G has a largest hypercube of dimension κ(G) as a subgraph, then σ(G) ≥ κ(G) (Proposition 5.1) 2 . In light of the second part of Huang's question it is thus natural to ask when this bound is tight. Clearly, all the three above families and also the family of [21] have κ ≡ 1 and hence they give tight examples for this bound. In [10], Chung, Füredi, Graham, and Seymour show that Huang's bound is tight for the hypercube itself, i.e., σ(Q d ) = √ d . We generalize this construction to sublattices of the hypercube (Lemma 5.3).
We obtain infinite families of Cayley graphs with unbounded κ, where Huang's lower bound is tight. Namely, we study Coxeter groups. Our main result here is that the Cayley graph G of a 1 Even if graphs are considered undirected, in figures we use arcs to represent generators of order larger than 2 to increase readability. 2 Note that this observation is also essential for the result for abelian groups in [26].
Coxeter group of type A n or I 2 (2k + 1) as well as their direct products satisfy σ(G) = κ(G) (Corollary 6.7). We furthermore extend this result to type I 2 (n) × I 2 (n ) (Theorem 6.8) as well as to many small Coxeter groups with the help of the computer (Table 6). Moreover, we show that graphs G of Coxeter groups of type B n and D n satisfy σ(G) ≤ κ(G) + 1 (Theorem 6.9). We conjecture, that every Cayley graphs G of a Coxeter group σ(G) = κ(G) (Conjecture 6.10). Next, we study the sensitivity of bipartite Cayley graphs in the absence of cubes, i.e., where Proposition 5.1 cannot be applied. We show that the Levi graphs of projective planes have unbounded sensitivity (Theorem 7.1). Thus, providing a family of cube-free, bipartite Cayley graphs that behave similarly to the hypercube with respect to sensitivity.
In the final section, after some concluding remarks we give an outlook on sensitivity in nonbipartite Cayley graphs. We show that the first guess on how to generalize the hypercube to higher chromatic number fails (Theorem 8.1).
The following result provides a family of bipartite (d + 1)-regular dihedrants with sensitivity 1 for all d ≥ 0.
Proof. Denote c = a 3 b for all 0 ≤ ≤ d. Take x ∈ M and let us prove that it has exactly one neighbor in M . We separate the proof in four cases.
If x = a i with [i] 3 = 1. We take j the largest exponent such that 3 j divides i and write i = j<m<d β m 3 m + 3 j . We observe that for all ∈ {0, . . . , d} Hence xc = a i a 3 b = a i+3 b ∈ M if and only if = j. As a consequence, x has exactly one neighbor in M .
We take j the largest exponent such that 3 j divides i and write i = j<m<d β m 3 m + 2 · 3 j . We observe that for all ∈ {0, . . . , d} Hence xc = a i ba 3 b = a i−3 ∈ M if and only if = j. As a consequence, x has exactly one neighbor in M .
Hence M induces a matching and it is easy to check that M has n + 1 elements. As a consequence σ(G) = 1.
Exhaustive enumeration by computer shows that there is no smaller bipartite non-cyclic Cayley graph with σ = 1, than the cubic 18-vertex dihedrant given by Theorem 2.1 for n = 9. This graph has been obtained earlier by [21].

STAR GRAPHS
The star graph is the bipartite graph SG n = Cay(S n , {(12), (13), . . . , (1n)}). As main result of this section, we will see in Theorem 3.3 that star graphs all have sensitivity equal to one. In other words, we will show that they have an induced subgraph with more than half of the vertices and maximum degree equal 1.
Lemma 3.1. Let n ∈ Z + and denote by d n the number of derangements of n elements. Then, d n is odd if and only if n is even.
Proof. It is easy to check that d n satisfies the recursive formula d n = (n − 1)(d n−1 + d n−2 ) for all n ≥ 3. Since d 1 = 0 and d 2 = 1, the result follows by induction. Proof. Since π and τ are adjacent in SG n , then τ = π · (1r) for some r ∈ {2, . . . , n}. We are going to prove that the symmetric difference of supp(π) and supp(τ ) is contained in {1, r} and, hence, the result follows. We write π = c 1 · · · c t as a product of cycles with disjoint support, we clearly have that supp(π) = ∪ t i=1 supp(c i ). We divide the proof in several cases: If 1, r / ∈ supp(π). Then τ = c 1 · · · c t · (1r) is a product of cycles with disjoint support, thus supp(τ ) = supp(π) ∪ {1, r}.
Proof. Let H be the domino, that is, the graph with vertices {u 1 , u 2 , u 3 , v 1 , v 2 , v 3 } and edges Let us check that f is a graph homomorphism (see Figure 1 for an example when n = 4). We observe that f (A n ) = {u 1 , v 2 , u 3 }, f (S n − A n ) = {v 1 , u 2 , v 3 }. Since the domino is the complete bipartite graph K 3,3 minus the edges u 1 v 3 , v 1 u 3 , in order to prove that f is a homomorphism we just have to check that if f (π) = u 1 and f (τ ) = v 3 (respectively, f (π) = v 1 and f (τ ) = u 3 ), then π and τ are not neighbors in SG n ; this follows from Lemma 3.2.   Now we are going to prove that the induced subgraphs K and K with vertices f −1 ({u 1 , u 2 , v 3 }) and f −1 ({v 1 , v 2 , u 3 }), respectively, have both maximum degree equal to 1. Let π ∈ V (K), we separate the proof in three cases: Case f (π) = v 3 . Then π has no neighbors in K (since f is a homomorphism).
A similar argument works for K . To get the result we are proving that K and K do not have the same number of elements and, as a consequence, one has more than half of the vertices of SG n (see Figure 1 for the case n = 4, where K has 13 vertices). Since f −1 ({u 1 , v 2 , u 3 }) = A n and f −1 ({v 1 , u 2 , v 3 }) = S n − A n and both sets have the same cardinality, we just need to verify that f −1 (u 1 ) and f −1 (v 1 ) do not have the same number of elements. It suffices to observe that the elements of f −1 ({u 1 , v 1 }) are in bijection with the set of derangements of either n or n − 1 elements and, thus, |f −1 (u 1 )| + |f −1 (v 1 )| = d n + d n−1 which, by Lemma 3.1, is an odd number. This completes the proof.
One can be more precise in the proof of Theorem 3.3 and determine that K has exactly n! 2 + (−1) n+1 vertices and K has n! 2 + (−1) n vertices. Indeed, following the notation of the proof, we have that |f −1 (u 1 )| equals the number of even (belonging to A n ) derangements of n elements plus the number of even derangements of n − 1 elements, then by [27, sequence A003221] we have that Thus, we conclude that the graph with more that half of the vertices of SG n is K for n odd, and K for n even.

TIGHT GROUPS
It is easy to see that an induced subgraph of maximum degree 1 in a d-regular n-vertex graph has at most d 2d−1 n vertices. We say that a graph is tight if it attains equality. Lehner and Verret ask if there are tight Cayley graphs of groups, see [21,Remark 2]. Here we give some examples and an infinite family.
First of all one can choose two generators b, c of D 3m of order 2 such that Cay(D 3m , {b, c}) ∼ = C 6m , the cycle graph on 6m vertices. This graph has an induced matching on 2 3 of the vertices, hence it is tight of degree 2.
An exhaustive computer search yields that on up to 60 vertices there are exactly three tight cubic Cayley graphs. Two of them on 50 and 60 vertices, respectively, are depicted in Figure 2. The other one is another Cayley graph of A 5 and is the first member of the infinite family shown in Theorem 4.1.
2m+1 |Γ| elements and induces a matching in Proof. We observe that the signature of c k is (−1) m and then c k ∈ A 2m+1 if and only if m is even. Now, we consider the partition of for all i, and then M has m+1 2m+1 |Γ| elements. Take i ∈ {m + 1, . . . , 2m + 1} and consider π ∈ M i . We claim that π · c k ∈ M if and only if k = i and, as a consequence, M induces a matching in G.
We wonder if the set C described in this result is a minimal set of generators of Γ in every case. Otherwise, the subgroup of Γ spanned by C would provide a smaller tight group.
It is also worth pointing out that the same result (and the same argument of the proof) holds for We remark that while the above construction gives a tight Cayley graph for every degree, the obtained graphs are pretty large. E.g., for degree 4, we obtain a Cayley graph of S 7 . However, we know of at least one smaller such graph, namely Cay(A 7 , {(1234567), (123)(45)(67)}). It has degree 4 and 2520 vertices and an induced matching of 1440 vertices, i.e., it is tight. We wonder what size the smallest 4-regular tight Cayley graph is. By computational means we checked that the answer is at least 84.
Note further, that the above graphs are the only non-bipartite graphs that have appeared so far. However, we can also construct bipartite ones. For this we recall a couple of definitions and prove a lemma that has been used implicitly in [21,26]. A covering map from a graphĜ to a graph G is a surjective graph homomorphism ϕ :Ĝ → G such that for every vertex v ∈Ĝ, ϕ induces a one-to-one correspondence between edges incident to v and edges incident to ϕ(v). If there is a covering map fromĜ to G, we say thatĜ is a covering of G. Finally, for a graph G and every Proof. Let ϕ :Ĝ → G be a covering map and let us assume without loss of generality that G is connected. It is easy to see that all fibers of ϕ have the same size k and, thus |V Since ϕ is a homomorphism and two neighbors of a given vertex cannot be mapped by ϕ to the same vertex, then the maximum degree induced by U is at most the maximum degree induced by ϕ(U ). This yields the claim.  The smallest bipartite tight Cayley graph that we are aware of comes from the above construction and is a Cayley graph of Z 10 × D 5 . With the computer we checked that the smallest cubic bipartite tight Cayley graph is of order at least 80. What is the smallest one?

BOUNDS AND CONSTRUCTIONS CLOSE TO THE HYPERCUBE
In the present section we give a very elementary generalization of the lower bound of Huang [20] and a more involved generalization of the construction of Chung, Füredi, Graham, and Seymour [10]. Both will be applied in the following section to Coxeter groups.
For any graph G, we denote by i.e., κ(G) is the dimension of the largest hypercube contained in G.
Proposition 5.1. Let G be a bipartite Cayley graph and H a subgraph of G, then σ(G) ≥ σ(H).
Proof. Since G and H are bipartite, their maximum independent sets contain half the vertices. Now, for every x ∈ V (G), we consider the set of vertices The sets (x · H) x∈V (G) cover G and every element in G belongs to exactly |V (H)| of these sets. If one and, by the pigeonhole principle, there exists an x ∈ V (G) such that |K ∩ (x · H)| > 1 2 |V (H)|. Since the induced graph with vertices x · H is isomorphic to H, we conclude that the maximum degree of the subgraph induced by K is at least σ(H). The second statement follows from Huang's result [20]. In [10], Chung, Füredi, Graham, and Seymour exhibited an induced subgraph of Q n with 2 n−1 + 1 vertices and maximum degree √ n for all n ≥ 1. Next we extend this construction to certain lattices.
We introduce some notation for posets and lattices. For a poset P we denote by G P = (P, E) its cover graph, i.e, {x, y} ∈ E if x < y and there is is no z ∈ P with x < z < y. We say that P ⊆ Q are cover subposets if x ≤ P y ⇐⇒ x ≤ Q y for all x, y ∈ P and G P is a subgraph of G Q . For x ∈ P denote by ↑ x = {y ∈ P | x ≤ y}. A lattice L is a partially order set, such that for any x, y ∈ L there a unique smallest element x ∨ y ≥ x, y called the join of x and y and a unique largest element x ∧ y ≤ x, y called the meet of x and y. The Boolean lattice B n is the inclusion order of all subsets of the set [n] = {1, . . . , n}.
Let L be a lattice that is a cover subposet of B n . Without loss of generality we identify the elements of L with sets such that the minimum and maximum0,1 of L correspond to the empty and the full set, respectively. We call the vertices even and odd depending on the cardinalities of the corresponding sets. The set of even and odd vertices of a subset S ⊆ L is denoted even(S) and odd(S), respectively. For F ⊆ L define r(F) = max{|F | | F ∈ F} and t(F) = max{|X| | X ⊆ F and ∀F ∈ X : F \ ( K∈X K =F K) = ∅}. This is, t(F) denotes the size of a largest subset X of F such that every F ∈ X contains an element that is in no other set from X. Given F we define: as the induced subgraph of G L on X(F) and on the complement of X(F), respectively. Lemma 5.3. Let L be lattice that is a cover subposet of B n , F ⊆ L, and k = max{r(F), t(F)}. We have: Proof. Since the statement for G (F) is proved analogously, here we only prove ∆(G(F)) ≤ max{r(F), t(F)}. So let {S, S } be an edge of G(F). If S is even, then S ≺ S is a cover relation, S is odd, and for all F ∈ ↓ (S) ∩ F we have S ∨F = S. Thus, the coordinate corresponding to the element S \S is contained in ( ↓(S)∩F).

Hence, deg(S) ≤ | ( ↓(S) ∩ F)| ≤ r(F).
If S is odd, then S ≺ S is a cover relation, S is even and the only element s ∈ S \ S belongs to some F ⊆ S = S ∪ {s} such that F ∈ F. Thus, the neighbors S ≺ S 1 , . . . S k give rise to a set For the second part of the statement we estimate |X(F)| = |even( ↑ F)| + |odd(L\ ↑ F)| via inclusion-exclusion. The size of the first term can be written as Similarly, we can express the size of the second term as: Since G L is bipartite and regular we have |odd(L)| = |even(L)| = n! 2 . We can write |X(F)| as:

COXETER GROUPS
We consider Cayley graphs of Coxeter groups and provide explicit constructions showing that the bound in Proposition 5.1 in each case is either an equality or at most one unit away from an equality.
More precisely, we first introduce the notion of cube-like Coxeter groups. This allows us to establish equality for Coxeter groups of types I 2 (2k + 1), A n , and their products. Further we show equality for types I 2 (n) and I 2 (n) × I 2 (n ), and many small Coxeter groups by computer. We also show that types B n and D n cannot deviate by more than one unit from the lower bound. We finish the section with a conjecture.
We start with the necessary definitions and refer the reader to [6,29] for more thorough introductions into the combinatorics of Coxeter groups. A finite Coxeter system is a pair (W, S), where W is a group with generators S = {a 1 , . . . , a n } and presentation W = a 1 , . . . , a n | (a i a j ) m ij = 1 where m ij > 1 and m ii = 2. In [11], Coxeter classified all finite Coxeter groups as (direct products of) the members of three infinite families of increasing rank A n , B n , D n , one family of dimension two I 2 (n) and six exceptional groups: E 6 , E 7 , E 8 , F 4 , H 3 and H 4 . Since any Coxeter group W corresponds to a unique Coxeter system (W, S), we denote the Cayley graph Cay(W, S) just as Cay(W ). See Figure 3 for a drawing of the Cayley graphs of A 3 and B 3 . It is well known that the Cayley graph of a Coxeter groups Cay(W ) is the dual graph of a simplicial hyperplane arrangement -the Coxeter arrangement of the corresponding type. Therefore Cay(W ) is an induced subgraph of the hypercube -even an isometric one, see e.g. [13]. The dimension of the hypercube corresponds to the number of hyperplanes in the arrangement, which is the number r of reflections. See Table 6 for the number of reflections of the irreducible Coxeter groups and for the dimension of the largest hypercube contained in their corresponding Cayley graphs, this number coincides with the size of the largest independent set of their Coxeter-Dynkin diagrams. The simpliciality of the Coxeter arrangement implies that Cay(W ) is the cover graph of a lattice, see [7]. Namely, taking as base-point of Cay(W ) the neutral element e ∈ W , the obtained lattice L W , is called the weak (right) order [5]. Note that with this we are already in the position to apply Lemma 5.3. We will however proceed to introduce more specific properties of L W , mostly taken from [6,29].
The length of an element w ∈ W is (w), which is the distance from e in Cay(W ). For J ⊆ S, we denote by W J the subgroup of W generated by J. The Coxeter system (W J , J) is called a parabolic subgroup of (W, S). Note that the graph Cay(W J ) is a subgraph of Cay(W ) and hence σ(Cay(W J )) ≤ σ(Cay(W )), by Proposition 5.1.
The set W J = {w ∈ W | (wj) > (w) for all j ∈ J} is the corresponding quotient. We collect some facts about W J and W J with respect to L W .
(1) W J is an interval I(W J ) and induces a sublattice of L W , (2) W J defines an interval I(W J ), whose graph we denote by G(W J ), moreover I(W J ) is isomorphic to the reversed order I(W J ) * , We call a Coxeter system (W, S) cube-like if it admits an abelian parabolic subgroup W J such that ι 0 (G(W J )) > 0. A consequence of Lemma 6.1 together with Theorem 5.4 is: Proposition 6.2. If (W, S) is cube-like with respect to J ⊆ S, then we have σ(Cay(W )) = κ(Cay(W )) and Cay(W ) has an induced subgraph of maximum degree σ(Cay(W )) on |W | 2 + ι 0 (G(W J )) vertices.
Proof. Denote d = κ(Cay(W )), by Proposition 5.1 we have that σ(Cay(W )) ≥ √ d . Since W J is abelian and minimally generated by J, Cay(W J ) is a cube contained in Cay(W ). Denote its dimension by d = |J|, we have that d ≤ d. By Theorem 5.4, we have that ι √ d Cay(W J ) ≥ 2. Lemma 6.1 yields As a consequence σ(Cay(W )) ≤ √ d ≤ √ d ≤ σ(Cay(W )); so they are all equalities and we are done.
A useful feature of cube-like Coxeter groups is that they are closed under products: Proof. If J, J yield the two parabolic subgroups witnessing that (W, S) and (W , S ) are cubelike, then also J × {e } ∪ {e} × J generates an abelian parabolic subgroup of (W × W , S × {e } ∪ {e}×S ). The graph G of quotient W ×W J×{e }∪{e}×J is the Cartesian product G(W J ) G(W J ). It follows from Lemma 5.2, that We present some necessary and one sufficient criterion for being cube-like: Proposition 6.4. Let (W, S) be a Coxeter system with r reflections. If (W, S) is cube-like with respect to J, then (1) κ(Cay(W )) = |J| , (2) r − |J| is even, (3) J is inclusion-maximal with respect to generating an abelian subgroup. Conversely, if r − |J| is even and the middle layer of I(W J ) is odd, then (W, S) is cube-like with respect to J.
Proof. 1. This is proved implicitly in Proposition 6.2. 2. The length of any shortest path from the minimum to the maximum of L W is r and for any parabolic subgroup W J , such path can be obtained by first going from the minimum to the maximum of L W J and then traverse a translate of the interval I(W J ). Since the diameter of the cube generated by J is |J| we get that r − |J| is the length of I(W J ). Now, we use the fact 2., that order reversing is an automorphism of I(W J ). If the length of I(W J ) is odd, then this automorphism identifies layers of different parity, hence both parts of a bipartition of G J will be of the same size and ι 0 (G J ) = 0. 3. Whenever W J is abelian for some J ⊆ S, then J corresponds to an independent set in the Coxeter-Dynkin diagram of W . When (W, S) is cube-like with respect to J ⊆ S, then J corresponds to a maximal independent set. Indeed, if this is not the case, there exists J J ⊆ S such that W J is abelian. As a consequence, G(W J ) = G(W J ) G(W J \J ) and G(W J \J ) = Q |J \J| , a hypercube of dimension |J \ J| ≥ 1. Hence, G(W J ) G(W J ) Q |J \J| ; but this implies that ι 0 (G(W J )) = 0, a contradiction.
For the sufficient condition, if we have an odd number of layers such that by fact 2. opposite ones are of the same size the bipartition class not containing the middle layer is even, but the one containing the middle layer will be odd. Hence ι 0 (G J ) > 0.
From Proposition 6.4 together with Table 6 we can infer that the the following Coxeter groups are not cube-like: B 2(n 2 +1) , B 2(n+1) 2 +1 for n even, D 8n 2 , D 8n 2 +1 for n ≥ 1, I 2 (n) for n = 0 mod 2 and E 6 . We will show next that Coxeter groups of type A n and I 2 (2k + 1) are cube-like. As a consequence any Cayley graph G of them or their products satisfies σ(G) = κ(G) . The Coxeter system I 2 (n) is (D n , {b, c}), where both b and c are generators of order 2.
Proof. The graph Cay(I 2 (2k + 1)) is a cycle of length 4k + 2. The maximal abelian parabolic subgroup is generated by a single element j, i.e., the maximal cube is an edge. The corresponding quotient I(W j ) is an interval consisting of a single chain of length 2k. In particular the middle layer is odd and ι 0 (G j ) > 1, by Proposition 6.4.
The symmetric group S n+1 with generators S = {(12), (23), . . . , (n(n + 1))} constitutes the Coxeter system of type A n . As an example consider A 3 . Its illustration in the left of Figure 3 shows that this Coxeter group is cube-like, even though it does not satisfy the sufficient condition in Proposition 6.4. This exemplifies the construction shown below. Theorem 6.6. For all n ≥ 0 the Coxeter system A n is cube-like with respect to a set J such that ι 0 (G(A n J )) ≥ n 2 !. Proof. We set J = {(12), (34), . . . , (nn + 1)} is n is odd and J = {(12), (34), . . . , (n − 1n)} otherwise. Clearly the parabolic subgroup generated by J is abelian. For the proof we identify the permutations with strings of length n + 1 in the standard way, e.g., e = [1, 2, 3, . . . , n + 1]. For a permutation π, its length (π) equals the number of pairs that are ordered differently from [1, 2, . . . , n + 1]. We refer to the two bipartition classes of Cay(A n ) as even and odd and correspondingly denote the parity of a permutation π by p(π) ∈ {0, 1}. Let P J be the poset on {1, . . . , n + 1} whose relations are of the form (i ≺ j) if (ij) ∈ J. Now W J can be seen as the set of linear extensions of P , i.e., all linear orders on {1, . . . , n + 1} that respect the relations prescribed by P .
Let us first consider the case n even. In this setting P consists of the single element {n + 1} and a disjoint union of chains 1 ≺ 2, . . . , n − 1 ≺ n called M . We label these chains C 1 , . . . , C n 2 . For the sake of the proof we say that an arc-diagram D is a perfect matching of K n . Any linear extension L M of M corresponds to an arc diagram, where each edge is labeled with a chain among C 1 , . . . , C n 2 . More precisely, a linear extension of M can be seen as a permutation π of {1, . . . , n} such that i < j whenever π(i) ≺ π(j); then the linear extension corresponds to the arc-diagram D with edges e j = (π(2j − 1), π(2j)) for 1 ≤ j ≤ n/2, and the edge e j is labeled by C j . Thus, given one arc diagram D there are n 2 ! linear extensions with this diagram. Moreover, all of them have the same parity p(D). To see the latter it is sufficient to distinguish how two arcs intersect whose assigned chains are exchanged. We skip this case distinction. Now, there are n + 1 possible ways to insert {n + 1} into a given linear extension of M with diagram D. Note that n+1 2 of these have parity p(D) and n+1 2 of these have parity (p(D) + 1) mod 2.
Since the number of arc-diagrams, i.e., the number of perfect matchings of K n is odd, for some p ∈ {0, 1} there is one more arc-diagram of parity p than there are of parity (p + 1) mod 2. So, take a diagram D of parity p. It corresponds to n+1 2 n 2 ! linear extensions of parity p and n+1 2 n 2 ! linear extensions of parity p + 1 mod 2. We thus have ι 0 (G J ) ≥ n 2 !. In the case that n is odd, the same proof works except that P is entirely partitioned into chains of length 2. The analogous analysis yields ι 0 (G J ) ≥ n+1 2 !.
The results of the present section can be applied to the sensitivity of some Coxeter groups: Corollary 6.7. Let G be the n-vertex Cayley graph of the product Then σ(G) = κ(G) and there exists a set of n 2 + Π j =1 ( n 2 !) vertices inducing this degree.
We proceed to study σ for Coxeter groups, where we cannot apply the above strategy.
Theorem 6.8. Let G be the Cayley graph of a Coxeter group of type I 2 (n) or I 2 (n) × I 2 (n ). Then σ(G) = κ(G) .
Proof. The Cayley graph of I 2 (2) is a square and, then, κ(I 2 (2)) = 2 and σ(I 2 (2)) = 2 = √ 2 . Let us see that also the product of two even cycles C i C j has a subgraph on more than half the vertices with max degree at most 2. If i = j = 4, one can take the subgraph consisting of an induced 8-cycle and a vertex without neighbors in this cycle. So assume that i > 4. Take a proper 3-coloring of C j with a, b, x, such that x is used at least once and such that the neighbors of every vertex colored x are colored differently. Now, in C i C j every copy of C i has color a, b or x. In every copy of C i colored with x, we always pick the same subgraph of maximum degree 1 and with more than i/2 vertices (we can do this because i > 4). In the other copies of C i we choose one of the two bipartition classes of C i , depending on whether its color is a of b. The resulting subgraph has more than half of the vertices and maximum degree 2.
The group D n is a subgroup of B n of index 2; it can be seen as the group with elements E [n] ×S n , where E [n] denotes the elements in 2 [n] with an even number of elements, and generators S = {a 1 , . . . , a n−1 , a n } with a i = (∅, (i i + 1)) for all i ∈ {1, . . . , n − 1} and a n = ({1, 2}, (12)) and .
Theorem 6.9. Let G be a Cayley graph of B n or D n . Then σ(G) ≤ κ(G) + 1.
We have shown that several Coxeter groups are tight with respect to the lower bound from Proposition 5.1. Also consider Table 6 (see also Figure 3 for B 3 ) for further results into this direction that were obtained by computer. Almost all the results from Table 6 have been obtained by solving a straight-forward integer linear program in CPLEX. The exception is E 6 . Here, an exhaustive search through all 2-elements sets J and the construction from Lemma 5.3 gave the desired set. Note in particular, while in every cube-like Coxeter group the construction from Lemma 5.3 yields a solution, E 6 is not cube-like by Proposition 6.4. For E 7 and E 8 even this method was not feasible by computer.
We believe to have gathered sufficient evidence to dare the following: Conjecture 6.10. Let G be the Cayley graph of a Coxeter group and Q d the largest subgraph isomorphic to a cube. Then G contains a set K of more than half the vertices, that induced a subgraph of maximum degree √ d , i.e., σ(G) = κ(G) .

THE ABSENCE OF CUBES
In a sense most of the paper so far has been about Huang's lower bound (Proposition 5.1) being tight, i.e., if a bipartite Cayley graph contains a largest cube Q d , then there is an induced subgraph of maximum degree at most √ d on more than half the vertices. However, we do not want to give the wrong impression that this lower bound is tight in general bipartite Cayley graphs. Denote by I q the Levi graph, i.e., point-line incidence graph, of the Desarguesian projective plane P (2, q). It is known that I q is bipartite and has girth 6. Moreover, I q is the Cayley graph of D q 2 +q+1 with respect to a set of q + 1 involutions, see [22,Theorem 1].
The following thus provides a family of cube-free (q + 1)-regular Cayley graphs and unbounded sensitivity.
Proof. Let us take a set P of p points and a set L of lines of P (2, q) such that p+ ≥ α(I q )+1 = |V (Iq)| 2 On the other hand, since the p − 1 sets of size at most t have to r-cover p points we have, (p − 1)t ≥ (p − 1)t ≥ rp and, hence, t ≥ rp/(p − 1). Substituting in the above equation we get and manipulating this expression we obtain (r − 1 2 ) 2 ≤ (p−1)(p−2) p + 1 4 and we get that r ≤ 1 2 + √ p Otherwise, if p > ( 1 2 q + √ q) 2 , the we have that Substituting in (1) and using the fact that ≥ p and, then, > (q 2 + q + 1)/2, we get that Note that we do not need the projective plane to be Desarguesian for the proof to work. Note also that at the proof of Theorem 7.1 as well as on Huang's result uses interlacing.
Using the list of vertex-transitive graphs on up to 47 vertices [8,19], we verified that each vertextransitive, bipartite G with girth at least 6 has σ(G) ≤ 2. Also compare sequences A185959 and A006800 in [27] for the numbers of Cayley and transitive graphs, respectively. In particular, examination by computer shows that σ(I q ) = 2 for q ≤ 4 and σ(I q ) = 3 for q = 5, 7. Indeed, I 2 is the well-know Heawood graph and yields the smallest transitive bipartite graph with girth 6 and σ = 2. The 62-vertex Levi graph of the Desarguesian projective plane P (2, 5) is the smallest transitive graph with σ = 3 that we know of. In particular, Theorem 7.1 yields σ(I 13 ) ≥ 4. We do not know of any bipartite Cayley graph of girth 6 with σ ≥ 4 apart from those provided by Theorem 7.1.
Note that the set of q + 1 involutions generating I q as a Cayley graph is not inclusion-minimal. Indeed, by [9, Theorem 1.1] the size of a largest minimal generating set of D q 2 +q+1 minus 1 is at most the size of a largest minimal independent set of Z q 2 +q+1 , which is at most the number of distinct primes dividing q 2 + q + 1, which is at most log(q 2 + q + 1) < q.
So while Theorem 7.1 gives rise to a family of Cayley graphs of bounded κ and unbounded σ one might still believe, that Cayley graphs with respect to minimal generating sets satisfy Huang's lower bound with tightness. However, the Möbius-Kantor graph G(8, 3) is bipartite and the Cayley graph Cay(P 1 , {X, Y, Z}), where and P 1 = {±I, ±iI, ±X, ±iX, ±Y, ±iY, ±Z, ±iZ} < SU (2) is the (first) Pauli group, see the left part of Figure 4. This group can also be described as central product of Z 4 with D 4 . While G(8, 3) has girth 6 one can check that σ(G(8, 3)) = 2 > 1 = κ(G (8, 3)) . Indeed G (8, 3) is also isomorphic to both Cay(M 16 , C) and Cay(QD 16 , C), where M 16 = {x r y s | x 8 = y 2 = e, yx = x 5 y} is the modular group of order 16, QD 16 = {x r y s | x 8 = y 2 = e, yx = x 3 y} is the quasidihedral group of order 16, and C = {x, y}. Yet another way of representing the Möbius-Kantor graph is as the dihedrant Cay(D 8 , {b, ab, a 3 b}). However, this generating set is not minimal. For further information on this remarkable graph, see [23].

CONCLUSIONS
Most of the paper is about Huang's lower bound (Proposition 5.1) being tight, i.e., if a bipartite Cayley graph contains a largest cube Q d , then there is an induced subgraph of maximum degree at most √ d on more than half the vertices. We show that this holds for some dihedrants, the star graphs, and some tight groups, where these results can be seen as proving insensitivity. We further prove the lower bound to be tight for large classes of Coxeter groups, and conjecture it for general Coxeter groups (Conjecture 6.10). On the other hand we show that there are also cubefree graphs of unbounded sensitivity -namely Levi graphs of projective planes. An important ingredient to this as well as to Huang's lower bounds is interlacing, which to a limited extent is a comment on Huang's second question. A curiosity is that the latter class as well as the first family of insensitive graphs are dihedrants with respect to non-minimal generating sets. While we have provided insensitive Cayley graph with respect to minimal generating sets, it remains open if there are bipartite Cayley graphs with respect to minimal generating sets that have bounded κ and unbounded σ (Question 7.2).
Further, we believe that the k-imbalance ι k , i.e., the coloring parameter associated to sensitivity σ deserves further investigation. Indeed, apart from cube-like Coxeter groups and in particular Q d , our results on star graphs and tight groups can be read in terms of this stronger parameter.
Let us finally conclude with some thoughts on non-bipartite Cayley graphs. Since many things already do not work in the bipartite case, let us go back to abelian groups. The result of [26] gives a lower bound on the induced maximum degree when more than half of the vertices are taken, but in a non-bipartite Cayley graph α is less than half the vertices. We show that the stronger version is false, i.e., there abelian groups with Cayley graphs of unbounded degree but σ ≡ 1. We do not know if there is a family of tripartite Cayley graphs playing the role of hypercubes with respect to σ.