Distance-2 MDS codes and latin colorings in the Doob graphs
aa r X i v : . [ m a t h . C O ] O c t Distance-2 MDS codes and latin coloringsin the Doob graphs ∗ Denis S. Krotov, Evgeny A. Bespalov
Abstract
The maximum independent sets in the Doob graphs D ( m, n ) areanalogs of the distance-2 MDS codes in Hamming graphs and of thelatin hypercubes. We prove the characterization of these sets statingthat every such set is semilinear or reducible. As related objects, westudy vertex sets with maximum cut (edge boundary) in D ( m, n ) andprove some facts on their structure. We show that the considered twoclasses (the maximum independent sets and the maximum-cut sets)can be defined as classes of completely regular sets with specified 2-by-2 quotient matrices. It is notable that for a set from the consideredclasses, the eigenvalues of the quotient matrix are the maximum andthe minimum eigenvalues of the graph. For D ( m, In this paper, we characterize the maximum independent sets in the Doobgraphs. The Doob graph D ( m, n ), as a distance-regular graph, has the sameparameters as the Hamming graph H (2 m + n, ∗ The work was funded by the Russian Science Foundation (grant No 14-11-00555). n = 3 (one of the coordinate isusually considered as dependent from the others). In nonassociative algebra,an n -ary quasigroup is exactly a pair of a set and an n -ary operation over thisset whose value table is a latin hypercube. Every maximum independent setin a Hamming graph is a completely regular code of radius 1; the nontrivialeigenvalue of this code is the minimum eigenvalue of the graph.For fixed q , the class of the distanse-2 MDS codes in H ( N, q ) is describedfor q ≤
4. For q ≤
3, the description is rather simple as for each n there isonly one such set, up to equivalence. In the case q = 4, there are 2 n + o ( N ) nonequivalent MDS codes in H ( N, q = 4 is a specialcase for the Hamming graphs H ( N, q ) from the point of view of algebraiccombinatorics: H ( N, N ≥ H ( N,
4) are the Doob graphs D ( m, n ), 2 m + n = N .The main goal of the current research is to extend the characterizationtheorem [7] for the distanse-2 MDS codes in H ( n,
4) = D (0 , n ) to the maxi-mum independent sets in D ( m, n ) with m >
0. The characterization theoremis formulated in Section 4 and proven in Section 6.As an intermediate result, in Section 5 we prove a connection betweenproperties of related objects, 2 × MDS codes, which can be defined as the setswith largest edge boundary (see Subsection 7.1). This result also generalizesits partial case m = 0, considered in [6].In Section 7, we consider alternative definitions of MDS and 2 × MDScodes in Doob graphs. In particular, we show that these two classes can becharacterized in terms of equitable 2-partitions with quotient matrices whosenontrivial eigenvalue is the minimum eigenvalue of the graph. In D ( m, × MDS codes (strictly speaking, the quotient matrix ofa new equitable partition is the arithmetic average of two quotient matri-ces corresponding to an MDS code and to a 2 × MDS code). The existence ofsuch “intermediate” objects between MDS and 2 × MDS codes is a new effect,2hich has no analogs in Hamming graphs.The next two sections contain preliminaries and auxiliary facts.
00 10 20 3001 11 21 3102 12 22 3203 13 23 33
Figure 1: The Shrikhande graph drown on a torus; the vertices are identifiedwith the elements of Z The
Shrikhande graph
Sh is the Cayley graph of the group Z with theconnecting set { , , , , , } (the vertices of the graph are the ele-ments of the group Z , which will be denoted 00, 01, 02, 03, 10, ..., 33; twovertices are adjacent if and only if their difference belong to the connectingset) see Fig. 1. The complete graph K = K of order 4 is the Cayley graph ofthe group Z with the connecting set { , , } . The Cartesian product of m copies of Sh and n copies of K will be denoted by D ( m, n ). This graph iscalled a Doob graph if m >
0, while D (0 , n ) is a 4-ary Hamming graph . Notethat D ( m, n ) is a Cayley graph of ( Z ) m × ( Z ) n , with the correspondingconnecting set.Given a graph G , by V G we denote its set of vertices. Two subsets of V G are said to be equivalent if there is a graph automorphism that maps onesubset to the other.An independent set of vertices of maximal cardinality, i.e. 4 m + n − , in D ( m, n ) is called a distance-2 MDS code (the independence number 4 m + n − = | V D ( m, n ) | / K , respectively). The two inequivalent MDS codes in D (1 ,
0) are shown inFig. 2.
Lemma 1.
If a subgraph G of D ( m, n ) is isomorphic to a Doob graph (withsmaller parameters), then the intersection of an MDS code with V G is anMDS code in G . f : V D ( m, n ) → V K is called a latin coloring if the preimageof every value is an MDS code; i.e., no two neighbour vertices have thesame colors. In the case m = 0, the latin colorings are known as the latinhypercubes (if n = 2, the latin squares ) of order 4. The case m = 1, n = 0 isillustrated in Fig. 3.The graph { ( y , ..., y m , x , ..., x n , x ) | x = f ( y , ..., y m , x , ..., x n ) } ofa latin coloring f is always an MDS code. Inversely, any MDS code in D ( m, n + 1) is a graph of a latin coloring of D ( m, n ) (however, the MDScodes in D ( m,
0) cannot be represented in such a manner). Moreover, if f and f ′ are latin colorings of D ( m, n ) and D ( m ′ , n ′ ), then the set { (¯ y, ¯ y ′ , ¯ x, ¯ x ′ ) ∈ V D ( m ′ + m, n ′ + n ) | f (¯ y, ¯ x ) = f ′ (¯ y ′ , ¯ x ′ ) } (1)is an MDS code in D ( m ′ + m, n ′ + n ). If 2 m + n > m ′ + n ′ >
1, thenthe MDS code (1) is called reducible , as well as all codes obtained from it bycoordinate permutation.A set M of vertices of D ( m, n ) is called a 2 × MDS code (two-fold MDScode) if every Shrikhande subgraph of D ( m, n ) intersects with M in 8 verticesthat form two disjoint MDS codes in Sh (see Fig 4) and every clique of order4 contains exactly 2 elements of M . A union of two disjoint MDS codes isalways a 2 × MDS code; such a 2 × MDS code will be called bipartite . Thecomplement V D ( m, n ) \ M of any 2 × MDS code M is also a 2 × MDS code,which will be denoted by M . However, it is not known if the complementof any bipartite 2 × MDS code is also a bipartite 2 × MDS code (the knownsolution [8] for D (0 , n ) is far from being easy). We will use this footenote mark to separate this notion from another meaning of theword “graph” (a pair of a set of vertises and a set of edges). × MDS codes in the Shrikhande graph, up to isomorphismA connected component of a subgraph of D ( m, n ) induced by a 2 × MDScode M will be called a component (of M ). A 2 × MDS code is called connected if it is a component itself. The union of (one or more) components of thesame 2 × MDS code will be referred to as a multicomponent .A 2 × MDS code is called decomposable ( indecomposable ) if its character-istic function can (cannot) be represented as a modulo-2 sum of two 0 , × MDS codes in Doob graphs,which, in their turns, can be decomposable or not. As a result, the charac-teristic function of any 2 × MDS code is a sum of the characteristic functionsof one, two, or more (up to m + n ) indecomposable 2 × MDS codes. It is easyto see that, with some natural assumptions, such a representation is unique:
Lemma 2.
The characteristic function χ M of any × MDS code M ⊂ V D ( m, n ) has a unique representation in the form χ M (¯ x, ¯ y ) ≡ χ M (˜ x , ˜ y ) + . . . + χ M k (˜ x k , ˜ y k ) + σ mod 2 (2) where • ¯ x = ( x , ..., x m ) ∈ Sh m ; • ¯ y = ( y , ..., y n ) ∈ K n ; • ˜ x i = ( x j , , ..., x j i,mi ) ∈ Sh m i , i = 1 , ..., k ; m = P k m i ; { , ..., m } = { j , , ..., j ,m , . . . , j k, , ..., j k,m k } ; • ˜ y j = ( y l i, , ..., y l i,ni ) ∈ K n i , i = 1 , ..., k ; n = P k n i ; { , ..., n } = { l , , ..., l ,n , . . . , l k, , ..., l k,n k } ; for all i ∈ { , ..., k } it holds m i + n i ≥ and M i is an indecomposable × MDS code not containing the all-zero tuple; • σ ∈ { , } . Obviously, an 2 × MDS code M is decomposable if and only if k > × MDS codes and semilinear MDScodes
We will say that a 2 × MDS code is linear if k = m + n in the representation(2), i.e., all M i are 2 × MDS codes in Sh or K , and, moreover, the ones thatare in Sh are not connected (Fig. 4(a)). It is easy to see that all linear2 × MDS codes in a given D ( m, n ) are equivalent.We will say that an MDS code is semilinear if it is a subset of a linear2 × MDS code. In the rest of this section, after auxiliary statements, weevaluate the number of semilinear MDS codes. We omit the proof of thenext lemma as it is obvious.
Lemma 3.
Let a characteristic function χ M of a decomposable × MDS code M has a decomposition χ M (¯ x, ¯ y ) ≡ χ M (˜ x , ˜ y ) + χ M (˜ x , ˜ y ) + σ mod 2 where each M i , i = 1 , , as well as its complement, has N i components. Then M , as well as its complement, has N N components. Lemma 4.
Let M be a × MDS code in D ( m, n ) , and let N be the numberof components in M .a) If M is a linear × MDS code then N = 2 m + n − .b) If M is bipartite, then the number of MDS codes included in M is N . Proof. a) The statement follows by induction from Lemma 3.b) As follows from the hypothesis, the subgraph D M of D ( m, n ) inducedby M is bipartite. By the definition of an MDS code, a part of D M is anMDS code. To choose a part of D M , one should independently choose one oftwo parts of every connected components of D M . So, the number of ways is2 N . N Lemma 5.
The number of linear × MDS codes in D ( m, n ) is · m + n . roof. Every linear 2 × MDS codes can be represented as (2) where foreach i ∈ { , , ..., k = m + n } the set M i is a 2 × MDS code in Sh equivalentto the one in Fig. 4(a) or a 2 × MDS code in K and, moreover, 00 M i . Inany case, M i can be chosen in 3 ways. Since σ can be chosen in 2 ways, wehave totally 2 · m + n ways to specify a linear 2 × MDS code. N Lemma 6.
The number of semilinear MDS codes in D ( m, n ) is · m + n · m + n − (1 + o (1)) as m + n → ∞ . Proof.
There are 2 · m + n linear 2 × MDS codes (Lemma 5), each includ-ing 2 m + n − semilinear MDS codes (Lemma 4). It remains to understand thatalmost every semilinear MDS code is included in only one linear 2 × MDScode. One of simple explanations of this fact is that two different linear2 × MDS codes M ′ , M ′′ can include at most one common MDS code. Indeedeach of M ′ , M ′′ is a coset of a subgroup of ( Z ) m × ( Z ) n , so the cardinality oftheir intersection cannot be larger than | M ′ | / | M ′′ | / m + n − , whichis the cardinality of an MDS code. N We are now ready to formulate the main result of the paper, which will beproven in the next two sections.
Theorem 1.
Every MDS code in D ( m, n ) is semilinear or reducible. Note that ‘or’ is not ‘xor’ here; a reducible MDS code (1) can also besemilinear. This happens when both graphs of f and f ′ are semilinear and,moreover, the representations (2) of the corresponding linear 2 × MDS codeshave the same summand χ M ∗ ( x ) corresponding to the dependent variable x ( M ∗ can be { (0 , , (1 , } , { (0 , , (1 , } , or { (1 , , (1 , } , but it is thesame for both f and f ′ ). Corollary 1.
The number of MDS codes in D ( m, n ) has the form · m + n · m + n − (1 + o (1)) as m + n → ∞ . Proof.
By Lemma 6, the number of semilinear MDS codes is 2 · m + n · m + n − (1 + o (1)). The number of reducible MDS codes is asymptoticallyinessential compairing with this value (see, e.g., [9]). N A key proposition
In this section, we will prove the following proposition, which will be used inthe proof of the main theorem.
Proposition 1. A × MDS code in D ( m, n ) , ( m, n ) = (1 , , is decomposableif and only if it is not connected. The following fact is easy to check directly.
Lemma 7.
Let M and M ′ be two × MDS codes in Sh . Then, either M and M ′ coincide, or they are are disjoint (i.e., M ∪ M ′ = V Sh ), or everycomponent of M intersects with every component of M ′ . The following partial case of Proposition 1 will be used as an auxiliarystatement.
Lemma 8. A × MDS code M in D ( m, n ) , m + n = 2 , is decomposable ifand only if it is not connected. Proof.
By Lemma 3 every decomposable 2 × MDS code is not connected,which is the ’only if’ part of the statement. For the ‘if’ part, we have toshow that any 2 × MDS code M in D ( m, n ), m + n = 2, is decomposable orconnected.The case of D (0 ,
2) = K × K is trivial, as there are only two nonequivalent2 × MDS codes, one is decomposable, the other is connected.The case of D (1 ,
1) = Sh × K is also simple. Denote M y = { x ∈ V Sh | ( x, y ) ∈ M } for each y ∈ V K . Then M y is a 2 × MDS code in Sh. If for all y ′ , y ′′ from V K the sets M y ′ and M y ′′ are either coinciding or disjoint, then,readily, M is decomposable. Assume that for some y ′ , y ′′ ∈ V K the sets M y ′ and M y ′′ are neither coinciding nor disjoint. By Lemma 7, the corresponding16 vertices of M belong to the same component. Moreover, for every y from V K , the set M y is neither coinciding nor disjoint with at least one of M y ′ , M y ′′ . It follows that M is connected.It remains to consider the case of D (2 ,
0) = Sh × Sh. For a ∈ Sh, denote R a = { x ∈ Sh | ( a, x ) ∈ M } and aR a = { ( a, x ) ∈ M } ,L a = { x ∈ Sh | ( x , a ) ∈ M } and L a a = { ( x , a ) ∈ M } . Consider the case when L is equivalent to the 2 × MDS code in Fig. 4(a)(the other two cases are considered similarly); without loss of generality,assume L = { , , , , , , , } . Now consider the component8 , , , } of L . If R = R = R = R , then for every a fromSh, the set { , , , } is a component of L a or L a ; hence, by Lemma 7, L a = L or L a = L . In this case, readily, M is decomposable. So, withoutloss of generality we can assume that R and R are different. Since theyintersect, Lemma 7 means that 00 R and 01 R lie in a one component of M . Since R and R intersect, 02 R lies in the same component; similarly,03 R . By similar arguments, each of S = 00 R ∪ R ∪ R ∪ R , S = 10 R ∪ R ∪ R ∪ R , S = 20 R ∪ R ∪ R ∪ R , S = 30 R ∪ R ∪ R ∪ R is a subset of a component of M , andit remains to show that this component is common for all the four sets. Forexample, let us show that it is common for S and S . Indeed, since R and R are different, R intersects with at least one of them; this means thatthere is an edge connecting 00 R ∪ R and 11 R . Similarly, there areedges connecting S and S , S and S . We conclude that M consists of onecomponent. N Lemma 9.
Let M be a × MDS code in D ( m, n ) , let L be its multicompo-nent, and let L be the set of vertices of D ( m, n ) at distance from L . Then L is a multicomponent of M . Proof.
Clearly, L is a subset of M . It remains to show that for each b ∈ L every c ∈ M adjacent to b also belongs to L . By the definition of L ,there is a ∈ L adjacent to b . Assume that a and b differ in the i th coordinate,while b and c differ in the j th coordinate.(*) We will show that there is d ∈ L adjacent to c and differing from c in the i th coordinate . If i = j is a K -coordinate, then this claim is obvious(we can take d = a ). If i = j is a Sh-coordinate, then it is also easy to see.Assume i = j and consider the subgraph D (isomorphic to D (0 , D (1 , D (2 , a , b , and c . Let M ′ be the intersection of M with this subgraph. We knowthat M ′ is a 2 × MDS code in D and hence, by Lemma 8, it is connectedor decomposable. In the first case, M ′ ⊆ L and (*) is trivial. In the lastcase, the vertex d coinciding with c in all coordinates except the i th one andcoinciding with a in the i th coordinate must belong to L (indeed, from thedecomposability we have χ M ′ ( a )+ χ M ′ ( b )+ χ M ′ ( c )+ χ M ′ ( d ) = 0 mod 2, hence d ∈ M ′ ; since d and a are adjacent, we also have d ∈ L ).So, (*) holds, and L consists of components of M . N Lemma 10.
Let M be a × MDS code in D ( m, n ) where m > . Let, for ome a , ..., a m ∈ V Sh and b , ..., b n ∈ V K , M ′ = { ( x , a , ..., a n , b , ..., b m ) ∈ M | x ∈ Sh } . Then either all elements of M ′ belong the same component of M or M isdecomposable and χ M ′ ( x ) + χ M ( x , ..., x n , y , ..., y m ) mod 2 (3) for some × MDS code M . Proof.
For α = ( α , ..., α m ) ∈ V Sh m − and ( β , ..., β n ) ∈ V K n , de-note M α,β = { x ∈ Sh | ( x , α , ..., α m , β , ..., β n ) ∈ M } . Clearly, M α,β is a 2 × MDS code in Sh. If M a,b is connected, where a = ( a , ..., a m ), b = ( b , ..., b n ), then, obviously, M ′ lies in a component of M and thestatement holds. Assume that M a,b is not connected and, moreover, M ′ intersects with two components of M . Without loss of generality, suppose00 M a,b . Define the 2 × MDS code M = { ( α, β ) | (00 , α, β ) M } ; inparticuar, ( a, b ) ∈ M . Our goal is now to prove (3).Let L be the component of M that contains ( a, b ). For every ( a ′ , b ′ ) ∈ M adjacent to ( a, b ) we have M a ′ ,b ′ = M a,b (otherwise, by Lemma 7, M ′ liesin one component, contradicting our assumption). Similarly, M a ′ ,b ′ = M a,b for every ( a ′ , b ′ ) from L . Define recursively, L i +1 as the set of vertices of D ( m − , n ) at distance 1 from L i , i = 0 , , ... . Since L is a component of M and M a ′ ,b ′ = M a,b for every ( a ′ , b ′ ) from L , we have M a ′′ ,b ′′ = M a,b for every( a ′′ , b ′′ ) from L . By Lemma 9, L is a multicomponent, and we can applythe same argument to get M a ′ ,b ′ = M a,b for every ( a ′ , b ′ ) from L . Similarly, M a ′ ,b ′ = M a,b for every ( a ′ , b ′ ) from L , L , L , ... and M a ′′ ,b ′′ = M a,b forevery ( a ′′ , b ′′ ) from L , L , L , .... It is easy to see that every vertex of M (of M ) belongs to L i for some even (odd, respectively) i . It follows that (3)holds. N Lemma 11 ([6, Corollary 4.2, Remark 4.3]).
Let G = K nq be theCartesian product of n copies of the complete graph K q of even order q .Let M be the set of vertices of G such that every clique of order q containsexactly q/ elements of M . Then either the subgraph G M of G induced by M is connected, or the characteristic function χ M of M is decomposable intothe sum χ M ( z ) = χ M ′ ( z ′ ) + χ M ′′ ( z ′′ ) mod 2 , where M ′ ⊂ V K n ′ q , M ′′ ⊂ V K n ′′ q ,and z ′ , z ′′ are nonempty disjoint collections of variables from z = ( z , ..., z n ) of length n ′ and n ′′ respectively, n ′ + n ′′ = n . emma 12. Let G ∗ = G ∗ ( m, n ) = K m × K n be the Cartesian product of m copies of the complete graph K of order and n copies of the completegraph K of order . Let M ∗ be the set of vertices of G ∗ such that everyclique K maximal by inclusion (it follows that | K | = 4 or | K | = 16 ) containsexactly | K | / elements of M . Then either the subgraph G ∗ M ∗ of G ∗ inducedby M ∗ is connected, or the characteristic function χ M ∗ of M ∗ is decomposableinto the sum χ M ∗ ( z ) = χ M ′ ( z ′ ) + χ M ′′ ( z ′′ ) mod 2 , where M ′ ⊂ V G ∗ ( m ′ , n ′ ) , M ′′ ⊂ V G ∗ ( m ′′ , n ′′ ) , and z ′ , z ′′ are nonempty disjoint collections of variablesfrom z = ( x , ..., x m , y , ..., y n ) . Proof.
We map each vertex ( x , ..., x m , y , ..., y n ) of G ∗ to 4 n elements( x , ..., x m , ( y , z ) , ..., ( y n , z n )), z i ∈ { , , , } , i = 1 , ..., n , which will betreated as vertices of K m + n . Then M ∗ is mapped into a set M of verticesof K m + n . It is easy to see that M satisfies the hypothesis of Lemma 11.Moreover, the subgraph G M is connected if and only if the subgraph G ∗ M ∗ isconnected; and the characteristic function χ M is decomposable if and only if χ M ∗ is decomposable. Then, the statement follows from Lemma 11. N Proof of Proposition 1.
Assume that M is a 2 × MDS code in D ( m, n ). If M is connected, then it is indecomposable by Lemma 3.Assume that M induces a disconnected subgraph of D ( m, n ). Since K m × K n is obtained from D ( m, n ) by adding some edges, the subgraph of K m × K n induced by M can be connected or not.If M induces a disconnected subgraph of K m × K n , then the statementfollows from Lemma 12.It remains to consider the case when the subgraph of K m × K n inducedby M is connected. This means that some Shrikhande subgraph of D ( m, n )intersects with two components of M (then, adding the absent edges in thissubgraph connects these two components). But then M is decomposable byLemma 10. N Proof of Theorem 1.
Consider an MDS code C in D ( m, n ), m > α = ( α , ..., α m ) ∈ V Sh m − and β = ( β , ..., β n ) ∈ V K n , denote C α,β = { x ∈ Sh | ( x , α , ..., α m , β , ..., β n ) ∈ C } .Assume that for all α and β the MDS code C α,β is equivalent to the codeshown in Fig. 2(a), i.e., one of C = { , , , } , C = { , , , } ,11 = { , , , } , C = { , , , } . Then, it is easy to see that C is reducible: C = { ( y, ¯ y ′ , ¯ x ′ ) ∈ D ( m, n ) | f ( y ) = f ′ ( y ′ , x ′ ) } for some f ′ : V D ( m − , n ) → { , , , } and for f : V D (1 , → { , , , } satisfying f ( y ) = ij for every y ∈ C ij .Otherwise, we may assume without loss of generality that C a,b = { , , , } (Fig. 2(b)) for some a and b . To utilize Proposition 1, we consider the 2 × MDScode M = C ∪ ( C + (01 , , ..., × MDS code M = { ( x , ..., x n , y , ..., y n ) | (00 , x , ..., x n , y , ..., y n ) ∈ C or (01 , x , ..., x n , y , ..., y n ) ∈ C } . In particular, ( a, b ) ∈ M . Similarlyto C α,β , we denote M α,β = { x ∈ Sh | ( x , α , ..., α m , β , ..., β n ) ∈ M } . Inparticular, M a,b = { , , , , , , , } (Fig. 4(a)).For every ( a ′ , b ′ ) ∈ M at distance 1 from ( a, b ), the MDS code C a ′ ,b ′ contains 01 and, moreover, is disjoint with C a,b = { , , , } . Thereis only one such code, C a,b = { , , , } . It follows that M a ′ ,b ′ = { , , , , , , , } = M a,b . Similarly, M α,β = M a,b for all ( α, β )from the same connected component of M as ( a, b ). Then, for all ( α, β )at distance 1 from this connected component, we have M α,β = M a,b . UsingLemma 9 and applying the same arguments as in the proof of Lemma 10,we see that for all ( α, β ) at even distance from the considered component wehave ( α, β ) ∈ M and M α,β = M a,b , while for odd distance, ( α, β ) M and M α,β = M a,b . It follows that χ M = χ M a,b ( x ) + χ M ( x ,...,x m ,y ,...,y n ) + 1 mod 2.So, we have that C is included in a decomposable 2 × MDS code M . Let(2) be the decomposition of M into indecomposable 2 × MDS codes M i , i =1 , ..., k , k ≥
2. By Proposition 1, all M i are connected, except some 2 × MDScodes in Sh, which are equivalent to the code in Fig 4(a). Assume without lossof generality that the first m ′ × MDS codes M , ..., M m ′ are not connectedand that they correspond to the first m ′ variables x , ..., x m ′ . Since M includes at least one MDS code C , all M i , i = 1 , ..., k are bipartite, as wellas their complements. It follows that for each i there is a latin coloring f i : V D ( n i , m i ) → V K such that M i = { z ∈ V D ( n i , m i ) | f i ( z ) ∈ { , }} .Define the linear 2 × MDS code L by χ L ( x , ..., x m ′ , z , ..., z k − m ′ ) = χ M ( x ) + . . . + χ M m ′ ( x m ′ )+ χ { , } ( z ) + . . . + χ { , } ( z k − m ′ ) + σ. It is straightforward that M = { (¯ x, ¯ y ) ∈ V D ( m, n ) | ( x , ..., x m ′ , f m ′ +1 (˜ x m ′ +1 , ˜ y m ′ +1 ) , ..., f k (˜ x k , ˜ y k )) ∈ L } . S ⊂ L , the MDS code { (¯ x, ¯ y ) ∈ V D ( m, n ) | ( x , ..., x m ′ , f m ′ +1 (˜ x m ′ +1 , ˜ y m ′ +1 ) , ..., f k (˜ x k , ˜ y k )) ∈ S } (4)is a subset of M . Since L and M consist of the same number of components,they include the same number of MDS codes. In particular, C is also rep-resentable in the form (4) for some S . If m ′ = m and k = m + n , then C is semilinear. Otherwise, k < m + n or k = m + n and m ′ < m , and C isreducible. N × MDS codes as sets withextremal properties
We already know that the MDS codes in a Doob graph are exactly themaximum (by cardinality) independent sets (actually, we take this propertyas the definition of the MDS codes). In this section, we show that the 2 × MDScodes also meet some extremal property. Namely, they are exactly the setswith maximum edge boundary (cut). Moreover, the MDS codes and the2 × MDS codes define equitable 2-partitions of the Doob graph with minimumeigenvalue.
The edge boundary (also known as cut ) of a set V of vertices of a graph G = ( V G, E G ) is the set of edges {{ v, w } ∈ E G | v ∈ V, w V } . Proposition 2. (a) The maximum size of the edge boundary of a vertex setin D ( m, n ) is (2 m + n )4 m + n . (b) A vertex set M has the maximum edgeboundary if and only if it is a × MDS code.
Proof.
It is straightforward from the definition that a 2 × MDS codehas the edge boundary of size (2 m + n )4 m + n . It remains to show the upperbound for the statement a) and the ‘only if’ part of b).(a) (2 m + n )4 m + n is 2 / D ( m, n ). Since D ( m, n ) is the Cartesian product of several copies of the Shrikhande graphSh and several copies of the complete graph K of order 4, it is sufficient toshow that the number of boundary edges in each of these two graphs cannotbe larger than 2 / ·
48 = 32 in the caseof Sh and · K . For K , this is trivial. For Sh, this follows from the13 Figure 5: There is a cycle of type 4:0 → Figure 6: There is a cycle of type 3:1fact that each triangle (complete subrgaph of order 3) cannot have more than2 boundary edges, while every edge belong to the same number of triangles.(b) We first note that it is sufficient to prove the statement for the graphsSh and K . Indeed, if we have a set M whose edge boundary size is 2 / D ( m, n ), then the same proportion takes plasein every Shrikhande or K subrgaph (otherwise, there is a contradiction withp.(a) for such subgraphs). If we already have the statement for Sh and K ,then M is a 2 × MDS code by the definition.For K , it is trivial. Let us consider the Shrikhande graph and a vertexset M with edge bound of size 32. We color the vertices of M by black andthe other vertices by grey. From the previous paragraph we know that eachtriangle has at least one black and one grey vertex; we will always keep thisfact in mind in the rest of the proof. The Shrikhande graph has 12 induced(i.e., without chords) 4-cycles. We consider three cases.(4:0) There is an induced cycle colored into one color. Then, the colorsof the other vertices are uniquely reconstructed (see Fig 5), leading to thesituation shown in Fig. 4(a).(3:1) There is a induced cycle with only one black vertex or only one greyvertex. The colors of six other vertices are uniquely reconstructed. Choosingthe color of any of the six remaining vertices uniquely leads to one of twomutually symmetric colorings, see Fig 6. The result corresponds to Fig. 4(b).(2:2) All induced cycles have two black and two grey vertices. Then, thereis an induced cycle containing two neighbour black vertices, see Fig 7. Fourvertices out of this cycle are uniquely colored. Choosing the color of anyof the eight other vertices and using only the hypothesis of the case (2:2),we uniquely color the seven remaining vertices, see Fig 7. Both M and its14 Figure 7: All cycles are of type 2:2complement are equivalent to the set shown in Fig. 4(c). N A partition ( P , ..., P r ) of the vertex set of a graph into r nonempty cells issaid to be an equitable partition , see e.g. [4, 9.3] (regular partition [1, 11.1B],partition design [2], perfect coloring [3]), if there is an r × r matrix ( s ij ) ni,j =1 (the quotient matrix ) such that for every i and j from 1 to r every vertexfrom P i has exactly s i,j neighbors from P j . It is known that each eigenvalueof the quotient matrix is an eigenvalue of the graph [1] (in a natural way, aneigenvector of the quotient matrix generates an eigenfunction of the graph).If the graph is regular, then its degree is always an eigenvalue of the quotientmatrix.The graph D ( m, n ) has the 2 m + n + 1 eigenvalues − m − n , − m − n + 4,. . . , 6 m + 3 n . In V D ( m, n ), we consider an equitable 2-partition ( C, C ) suchthat the quotient matrix has two eigenvalues − m − n and 6 m + 3 n (thesmallest and the largest eigenvalues of D ( m, n )). This matrix has the form (cid:2) aa + d d − a d − a (cid:3) for some a from 0 to 2 d . The cases a = 0, a = d , and a = 2 d correspond to C being an M DS code, a 2 × MDS code, and the complementof an
M DS code.
Proposition 3. a) A -partition ( C, C ) of V D ( m, n ) is equitable with thequotient matrix (cid:2) m + n m +3 n m +2 n (cid:3) (or (cid:2) m +2 n m +3 n m + n (cid:3) ) if and only if C (respec-tively, C ) is an M DS code. b) A -partition ( M, M ) of V D ( m, n ) is equitablewith the quotient matrix (cid:2) m + n m +2 n m +2 n m + n (cid:3) if and only if M is a × MDS code.
Proof.
We will use the formula | S | = cb + c | V G | for the cardinality of thefirst cell S of an equitable partition of a graph G with the quotient matrix (cid:2) ac bd (cid:3) .a) For the first matrix, we have a = 0; so, C is an independent set.Counting its cardinality gives the cardinality of an MDS code. For the secondmatrix, similar argument works for C .15igure 8: An equitable 2-partition with quotient matrix (cid:2)
13 53 (cid:3) b) Count the size | M | · b of the edge boundary and apply Proposition 2. N However, the cases a = 0, a = d , and a = 2 d are not all possible cases.In D ( m,
0) (i.e., d = 2 m ), the cases a = 0 . d and a = 1 . d are also feasible;that is there exists an equitable partition with the quotient matrix (cid:2) m m m m (cid:3) .The first ( m = 1) such partition is shown in Fig. 8. Proposition 4.
Let ( C, C ) be the partition shown in Fig. 8. Then (cid:16)n ( x , ..., x n ) ∈ V D ( n, (cid:12)(cid:12)(cid:12) n X i =1 x i ∈ C o , n ( x , ..., x n ) ∈ V D ( n, (cid:12)(cid:12)(cid:12) n X i =1 x i ∈ C o(cid:17) is an equitable partition with the quotient matrix (cid:2) m m m m (cid:3) . The proof is straightforward.
Remark.
We breafly discuss another important concept related to the equi-table partitions. A set S of vertices of a graph is said to be completely regular (often, a completely regular code ) if the partition of the graph vertices withrespect to the distance from S is equitable (the quotient matrix is tridiagonalin this case). The number of cells different from S in this partition is calledthe covering radius of S . Trivially, each cell of an equitable 2-partition is acompletely regular set of covering radius 1. As shown in [5], using the Carte-sian product, from every completely regular code of covering radius 1 one canobtain a completely regular code of arbitrary covering redius. It is interestingthat a component of a decomposable 2 × MDS code can be represented as theCartesian product of k × MDS codes, where k is from (2). As follows fromthe results of [5], such a component is a completely regular code, if and onlyif all 2 × MDS codes M i , i = 1 , ..., k in the decomposition (2) have the samequotient matrices. This happens if and only if 2 m + n = . . . = 2 m k + n k .16 Conclusion
We have proven a characterization of the distance-2 MDS codes (maximumindependent sets) in the Doob graphs D ( m, n ).For related objects in D ( m, n ), called the 2 × MDS codes, we have proventhe equivalence between the connectedness and the indecomposability. How-ever, the problem of characterization of the 2 × MDS codes (namely, thoseof them that cannot be split into two MDS codes) remains open. We haveshown that the 2 × MDS codes are exactly the sets with maximum cut (edgeboundary).We have noted that the MDS codes and the 2 × MDS codes in D ( m, n ) arethe equitable 2-partitions with certain quotient matrices. The eigenvalues ofthese matrices are the minimum and the maximum eigenvalues of D ( m, n ).We have found that in the case n = 0 there is a third class of equitable 2-partitions that corresponds to these eigenvalues. Characterizing all partitionsfrom this class is also an interesting direction for further research. The work was funded by the Russian Science Foundation (grant No 14-11-00555).
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