Claude Tardif
Non-canonical Independent sets in Graph Powers
Let s ≥ 4 be an integer. The truncated s-simplex Ts is defined as follows:
V (Ts ) = {(i, j) ∈ {0, 1, . . . , s − 1}2 : i 6= j},
E(Ts ) = {[(i, j), (k, l)] : i = k, j 6= l or i = l, j = k}.
The truncated simplices are vertex-transitive, and it is known that their
categorical powers have maximum independent sets that are non-canonical,
in the sense that they are not inverse images of an independent set under a
projection.
Question: Are there other families of vertex-transitive graphs whose powers
have non-canonical maximum independent sets? In particular, the core of Ts
is the clique Ks−1 (by Brook’s theorem), so families of core examples would
be interesting.
In contrast, recent research on Fourier analysis and spectral techniques
has outlined sufficient conditions for a graph to have only canonical maximum
independent sets in its powers.
Ameera Chowdhury
[email protected]
A Vector Space Analog of Lovasz’s Version of the Kruskal-Katona
Theorem
Let V denote a n-dimensional
vector space over a finite field of order q.
V +
For k ∈ Z , we write k q to denote the family of all k-dimensional subspaces
of V . For a ∈ R and k ∈ Z+ , define the Gaussian binomial coefficient by
Y q a−i − 1
a
:=
.
k−i − 1
k q
q
0≤i<k
If k and q are fixed, then ka q is a continuous function of a and is positive
and strictly increasing when a ≥ k. From now on, we will omit the subscript
q.
We define the shadow of F ⊂ Vk , denoted ∂F, to consist of those (k −1)dimensional subspaces of V contained in at least one member of F; that
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V is, ∂F := E ∈ k−1
: E ⊂ F ∈ F . Balász Patkós and I recently proved
the following vector space analog of Lovasz’s version of the Kruskal-Katona
theorem.
Theorem 1. Suppose F ⊂ Vk . Let y ≥ k be the real number defined by
y |F| = ky . Then |∂F| ≥ k−1
. If equality holds, then y ∈ Z+ and F = Yk ,
where Y is a y-dimensional subspace of V .
In extremal set theory, Lovasz’s version of the Kruskal-Katona theorem
has many applications. For example, Daykin essentially used Lovasz’s theorem to give a proof of the Erdős-Ko-Rado theorem. We willTcall a family
F ⊂ Vk r-wise intersecting if for all F1 , . . . , Fr ∈ F we have ri=1 Fi 6= {0}.
Balász Patkos and I applied Theorem 1 to give an upper bound on r-wise
intersecting families in vector spaces.
Theorem 2. Suppose F ⊂ Vk is r-wise intersecting and rk ≤ (r−1)n. Then
|F| ≤ n−1
. Moreover, equality holds if and only if F = F ∈ Vk : v ⊂ F
k−1
for some one-dimensional subspace v ⊂ V , unless r = 2 and n = 2k.
Observe that the case r = 2 of Theorem 2 is the Erdős-Ko-Rado theorem
for vector spaces. In the language of graph theory, the Erdős-Ko-Rado theorem for vector spaces gives the size and structure of a maximum coclique
in the so-called q-Kneser graph; the q-Kneser graph has the k-dimensional
subspaces of V as its vertices, where two subspaces α, β are adjacent if
α ∩ β = {0}.
It would be nice to discover new applications of Theorem 1. For example,
does Theorem 1 imply anything about t-intersecting or cross-intersecting
families in vector spaces?
Cheng Yeaw Ku
Bounds on the Eigenvalues of the Derangement Graph
Let Γn denote the Cayley graph on the symmetric group Sn generated by
the set Dn of derangements (fixed-point free elements). It is well known that
the eigenvalues of Γn are integers given by
ηχ =
1 X
χ(s),
χ(1) s∈D
n
2
where χ ranges over all the irreducible characters of Sn . Moreover, the irreducible characters of Sn are indexed by partitions λ of n. We write ηλ to
denote the eigenvalue ηχλ of Γn , where χλ is the irreducible character indexed
by the partition λ ` n.
Conjecture 1. [1] Suppose λ∗ ` n is the largest partition in lexicographic
order among all the partitions with λ1 as their first part. Then, for every
λ = (λ1 , . . . , λs ) ` n,
|η(λ1 ,1n−λ1 ) | ≤ |ηλ | ≤ |ηλ∗ |.
References
[1] C. Y. Ku and D. B. Wales, The eigenvalues of the derangement graph,
Journal of Combinatorial Theory Series A , article in press.
Reza Naserasr
A Conjecture of Alon-Saks-Seymour:
Conjecture 2. For a graph G, if the edges of G can be partitioned into n
complete bipartite graphs then G is n + 1 colourable.
We have concrete examples that might be counterexamples to this.
Bill Martin
Delsarte’s LP bound v.s. Hoffman bound
Consider a symmetric association scheme with associate matrices {A0 , A1 , A2 , . . .}.
When is the optimum solution to Delsarte’s LP for a {1, 2}-code better than
the Hoffman bound?
Uniform one-factorizations of the complete bipartite graph
Let {R[1n ] , Rλ1 , Rλ2 , ..., R[n] } (where [1n ], λ1 , λ2 , ..., [n] are all the integer
partitions of n) be the group association scheme (also called the conjugacy
class scheme) of the symmetric group Sn . For which integer partitions λ of
n, does there exist a clique of size n in the graph Rλ ?
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This was a “social” research project of mine back in the mid-90s, with
anyone interested welcome to make contributions. Gillian Nonay and I were
the main players, but there were contributions from Dan Archdeacon, Gordon
Royle and others. I have a web page on the subject (http://users.wpi.edu/ martin/RESEARCH/onefac.html). The case σ = [n] is well-studied in graph theory: these are called ”perfect one-factorizations” – the union of any two of
the matchings is a Hamilton cycle. Linear programming seems to be vacuous
in this situation, so other non-existence results are needed.
Karen Meagher
[email protected]
EKR Theorem for Partitions
Let P (k, k) denote the collection of all set partitions of {1, . . . , k 2 } into
k parts, each part of size k. Two partitions P and Q from P (k, k) are
“intersecting” if there is a part of P and a part of Q such that the size of the
intersection of these two part is at least 2. What is the largest collection of
partitions from P (k, k) such that any two partitions from the collection are
intersecting.?
There are obvious candidates for the largest such collection. Let i, j be
distinct values in {1, . . . , k 2 }. Let Si,j be the collection of all partitions from
P (k, k) that have a part that contains both i and j.
This problem can be rephased as a question about the maximum independent sets in a graph. Consider a graph whose vertex set is the set of
partitions from P (k, k) and vertices are adjacent if and only if they are skew,
that is, any two cells have intersection of size exactly 1. Call this the partition
graph.
An independent set in the partition graph is a collection of interseting
partitions. If k is a prime power then there is a clique of size k + 1 in the
partition graph (such a clique is equivalent to an orthogonal array). In this
case, from the clique/coclique bound we know that the sets Si,j are maximum
cliques. It would be nice to show that this is true for values of k and further to
show that the sets Si,j are all the maximum independent sets in the partition
graph.
Another interesting problem is to find the least eigenvalue for the partition
k−1
graph. It is not difficult to show that − k! k is an eigenvalue of the partition
graph. In the case where k is a prime power, from the ratio bound for
k−1
cliques, we know that − k! k is the least eigenvalue. I conjecture that this is
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the least eigenvalue for all values of k. If it can be shown that this is the least
eignevalue, then using the ratio bound for independent set, the collections
Si,j must be the largest independent sets in the partition graph.
Two final questions are: what is the chromatic number of the partition
graph and is the partition graph a core?
Induced Bipartite Subgraphs in the Kneser Graph
What is the largest induced bipartite subgraph in the Kneser graph graph
K(2k + 1, k) (or more generally K(n, k))?
It is possible to construct an induced bipartite subgraph by taking one
bipartition to be all the subsets that contain 1 and the other bipartition to
be all the subsets that contain 2, but not 1. But, in the graph K(2k + 1, k),
it is not hard to construct a larger induced bipartite subgraph. For example,
if k = 4 the size of the bipartite subgraph induced by the collection of sets
that contain 1 and the sets that contain the element 2 but not 1 is
8
7
+
= 91.
3
3
Next let A be the collection of sets in V (K(9, 4)) that have at least 2 of the 3
elements {1, 2, 3} and let B be the collection of sets in V (K(9, 4)) that have
at least 3 of the 5 elements {5, 6, 7, 8, 9}. The set A and B induce a bipartite
subgraph of K(9, 4) with size
3 6
3 6
5 4
5 4
+
+
+
= 96.
3 1
2 2
4 0
3 1
This induced subgraph is conjectured to be the largest. With the interia
bound it can be shown that the size of the largest induced subgraph in K(9, 4)
is no larger than 98.
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