the possible choices for the images of other vertices), and then turns to the next vertex, until either a full endomorphism is found, or there are unmapped vertices for which no possible choice of image respects the property that edges are mapped to edges.
For this reason we were able to determine all endomorphism monoid for the graphs with strictly less than45vertices; however, the correct sizes of the endomorphism monoids are still unknown for a few graphs on45and49vertices.
95
Chapter 5
Endomorphisms of Hamming Graphs
and Related Graphs
This chapter analyses and determines singular endomorphisms of graphs coming from the Hamming association scheme and various related graphs. The graphs arising from the Hamming scheme are of the following form: Letm ≥2,n ≥3and letSbe a proper subset of{1, ..., m}, and consider the graphΓwith vertex setZmn where two vertices are adjacent if their Hamming distance is inS. The automorphism group of this graph is the wreath productSnoSm with the primitive product action and permutation rankm+ 1.
However, the graphs related to this construction are graphs over hypercuboids and other graphs arising from Cartesian and categorical products.
Although in the literature it is common to speak of the unique Hamming graph (or rather a family of graphs), in this thesis all graphs coming from the Hamming association scheme are called Hamming graphs and are denoted by H(m, n;S). If S consists of a single element k, then we write H(m, n;k), and ifk = 1 we write simply H(m, n)
(which is the Hamming graph). Note that the complement graphH(m, n;S)is the graph
H(m, n;{1, ..., m} \S).
The aim of this chapter is to investigate the singular endomorphisms of these graphs for various sets S. The first two sections establish that, if S is one of {1},{2, ..., m},
{1, ..., m−1}or {m}, then all singular endomorphism are uniform. (Note it is known that singular endomorphisms exist). Subsequently, we count the endomorphisms. Then in Section 5.5, we generalise the results on the endomorphisms of the Hamming graph to H(m, n;S), for S = {1, ..., k} for some k, and to the cuboidal Hamming graph in Section 5.6.
The Hamming Graph and Its Complement
In some literature, the Hamming graph is the distance-transitive graph which is given by the Cartesian product ofmcopies of the complete graphKn:
Kn· · ·Kn.
However, it is usually defined as the graphH(m, n;S), for S = {1}, which is a more common description, and thus, we write H(m, n). Many results are known about the Hamming graph and its simple structure has been inspiring mathematicians for a long time; however, a description of its endomorphisms is missing.
In [37], the usual approach of finding endomorphisms was to determine the maximal cliques, check whether the necessary condition (ω·α=n) from Lemma 3.3.6 is satisfied, and then describe the action of an endomorphism on the cliques. For H(m, n) it is straightforward to see that the maximal cliques are lines. So in this section, it is shown that the singular endomorphisms of the Hamming graphH(m, n)are uniform of ranknk,
where1≤k ≤m−1, and that its complement graph is a pseudo-core. In a later section, the singular endomorphisms are described in more detail.
Before moving on, it is necessary to introduce new notation. From school everyone knows that one can draw a cube by drawing its layers iteratively. That is, a cube is a collection of two dimensional layers, which are squares. This concept applies to higher dimensions and is described here.
5.1. The Hamming Graph and Its Complement 97
be split into layers; in particular, it is possible to divide it inton layers with respect to, say, the first coordinatex1 by denoting theith layer by the set
li ={(i, x2, ..., xm) :x2, ..., xm ∈Zn}.
Each layer is of dimension m − 1 and can be subdivided into layers of dimension
m− 2, m −3 and so on. A k-layer denotes a k-dimensional layer, that is a coset of
nkpoints wherem−kcoordinates are fixed. Thus, alayer systemofk-layers (ork-layer systemis the set of allnm−kdisjointk-layers which add up toZmn.
Example 5.1.1. 1. A squareZ2
3 is a collection of3rows. Those rows form a1-layer
system. (1,1) (1,2) (1,3) (2,1) (2,2) (3,3) (3,1) (3,2) (3,3)
2. A cubeZ33 can regarded as the2-layer system
li ={(x1, x2, i) :x1, x2 ∈Zn}, fori= 1,2,3.
or as the1-layer system
lij ={(x1, i, j) :x1 ∈Zn}, fori, j = 1,2,3.
InH(m, n), the layer systems play an important role and so does the number of k- dimensional layers. Let hk(m, n) denote this number. The square lattice graph L2(n)
is the Hamming graphH(2, n); so recall from Section 4.1.1, for this graph the1-layers are the maximal cliques and their number is 2n. Also, there are certainly mn layers of dimension m − 1 in H(m, n). So this number is simply given by considering the coordinates. To obtain a k-layer, we need to choose k of the m coordinates, and for each such choice the remainingm−k coordinates which are fixed, but freely chosen. It
follows: hk(m, n) = m k nm−k.
Applying this formula to the number of maximal cliques inH(m, n), which in fact are1-layers, reveals that there areh1(m, n) =mnm−1 of them. Also, for(m−1)-layers
we havehm−1(m, n) =mn.
Endomorphisms of the Hamming Graph
In dimension m = 2 the Hamming graph H(2, n) is the square lattice graph L2(n),
and in Chapter 4 it was pointed out that its singular endomorphisms are Latin squares. Similarly, in this chapter it is shown that the singular endomorphisms of H(m, n) are Latin hypercubes.
First, note that in [17] Cameron has already established thatH(m, n)admits singular endomorphisms of ranksnk, for1≤k ≤m−1. Supporting this result, here it is proved
that these are the only ranks which occur. Moreover, the result in this section answers the question on whether or not the Hamming graph admits any non-uniform endomorphisms. The answer is - No!
Theorem 5.1.2. A singular endomorphism ofH(m, n) is uniform of ranknk, for some
1≤k ≤m−1, and its image is a layer of dimensionk.
This theorem is a consequence of the following lemma.
Lemma 5.1.3. Let φ be a singular endomorphism of H(m, n), and let l be a k-layer. Thenlφis a layer of dimensiond, where1≤d≤k.
Proof. We will use induction onmandk. LetA(m, k)be the hypothesis. The hypothesis is satisfied forA(2,1), A(2,2)(seeL2(n)in Theorem 4.1.1) andA(m,1)(an endomor-
phism maps maximal cliques to maximal cliques). We assume that the hypothesis holds forA(m, k)and show it holds forA(m, k+ 1).
5.1. The Hamming Graph and Its Complement 99
l
1l
2l
1φ
l
2φ
c
1c
2φ
c
1φ
c
2φ
Figure 5.1: Impossible configuration: c1φis not a clique anymore.
Let l be a (k + 1)-layer. Then, we can split l into parallel k-layers l1, ..., ln. By
inductionliφis ak-layer or a layer of smaller dimension, for alli. Now, if the dimensions
of, say,l1φandl2φwould differ, then there would be two maximal cliques (lines)c1and
c2 connectingl1 and l2 such that at least one of c1φ and c2φ would not be a line in the
image ofφ(cf. Figure 5.1). A contradiction. Therefore, allliφhave the same dimension,
sayd.
Using the same argument, we see that each li is collapsed to d-layer such that the
union of all thed-layersliφforms a(d+ 1)-layer. Thus, the imagelφis a(d+ 1)-layer.
Note eachli is collapsed to liφ uniformly; otherwise, by essentially the same argument
we would be able to find a maximal clique which is not mapped to a maximal clique.
Proof of Thm. 5.1.2. Letφbe a singular endomorphism and let lbe the whole m-layer. By the previous lemmal is ak-layer where1≤k < m.
Corollary 5.1.4. For any singular endomorphism φthere is a maximal numberk, such thatφmapsk-dimensional layers to1-dimensional layers.
The following should be clear.
Lemma 5.1.5.If a singular endomorphismφofH(m, n)collapses ak-dimensional layer
The Complement of the Hamming Graph
The complement ofH(m, n)is the graph H(m, n;S), where S = {2, ..., m}, and two vertices are adjacent if their Hamming distance is not1. Form = 2this is the complement of the square lattice graph which has been covered in the previous chapter; so here, we focus on higher dimensions. Recall from Section 4.1.1 that a maximal clique inH(2, n)
is of the form{(ig, i) : i = 1, ..., n} for a permutationg ∈ Sn, and when considering
these as1-dimensional Latin rows, then the next result says that the maximal cliques of
H(m, n)form Latin hypercubes.
Theorem 5.1.6. The maximal cliques inH(m, n)are in1−1correspondence with Latin hypercubes of dimensionm−1and ordern(and class1).
Proof. First, we note that a Latin hypercube is a maximal clique of sizenm−1. Hence, the clique number isnm−1. We use induction onm. The casem = 2is clear, so letCbe a maximal clique inH(m, n), form >2. Pick a layer systemliof(m−1)-dimensional
layers, fori = 1, ..., n. Each layer is a subgraph isomorphic to H(m−1, n), so it has clique number nm−2. Moreover, each layer contains exactly nm−2 vertices of C, since
otherwise, if there would be one layer containing at least nm−2 + 1 vertices of C, it
would have a maximal clique of sizenm−2 + 1, contradicting the induction hypothesis.
Therefore, the intersection C ∩li is a maximal clique for H(m−1, n) and has nm−2
vertices. Intersecting C with all possible layers of dimension m − 1, determines the coordinates of the vertices ofCand it turns out thatCis a Latin hypercube of dimension
m−1.
Theorem 5.1.7. The graphH(m, n)is a pseudo-core, i.e., all singular endomorphisms have ranknm−1and are uniform.
Proof. To prove this theorem, we make use of the same method as form= 2in Chapter 4. Letc1 and c2 be two distinct maximal cliques which are identified byφ, say, c1φ =