Binary covering arrays on tournaments

We introduce graph-dependent covering arrays which generalize covering arrays on graphs, introduced by Meagher and Stevens (2005), and graph-dependent partition systems, studied by Gargano, Körner, and Vaccaro (1994). A covering array CA(n; 2, G,H) (of strength 2) on column graph G and alphabet graph H is an n× |V (G)| array with symbols V (H) such that for every arc ij ∈ E(G) and for every arc ab ∈ E(H), there exists a row ~r = (r1, . . . , r|V (G)|) such that (ri, rj) = (a, b). We prove bounds on n when G is a tournament graph and E(H) consists of the edge (0, 1), which corresponds to a directed version of Sperner’s 1928 theorem. For two infinite families of column graphs, transitive and so-called circular tournaments, we give constructions of covering arrays which are optimal infinitely often. Mathematics Subject Classifications: 05D05, 05B30


Introduction
Covering arrays are generalizations of orthogonal arrays that have been largely studied for their combinatorial interest and applications to software and network testing (see [1] and references therein).
Meagher and Stevens [9] extended the definition of a covering array to include a graph structure related to its columns, which we call a column graph.While covering arrays optimize the number of tests required to test all pairwise interactions between parameters of a system, the graph structure introduced in [9] is used to specify pairs of parameters that are known not to interact, yielding further reductions in the number of tests required.In addition to this increase in efficiency, their paper provides interesting connections between graph homomorphisms and Sperner systems.
Tracing back to the work of Gargano, Körner, and Vaccaro [2,3] on Sperner capacities, another generalization of covering arrays, different from that introduced in [9], is implicitly considered.A graph (which we call an alphabet graph to distinguish it from the column graphs considered in [9]) is used to encode pairs of symbols which must appear in every two distinct columns of the array.Using the terminology given in [3], classical covering arrays of strength 2 correspond to "graph-dependent partition systems", where the alphabet graph being considered is complete with a loop on every vertex.While their work includes the case of classical covering arrays, it also implicitly defines another generalization of covering arrays, where the pairwise coverage requirement is restricted to pairs of symbols specified by the edges of the alphabet graph.
In this paper, we unify both types of graphs associated with covering arrays, generalizing the definition of covering arrays to graph-dependent covering arrays.Like in [9], the binary alphabet case provides interesting connections with extremal set theory.We also consider the more general case of directed alphabet graphs.
For a positive integer n, we write [n] to denote the set of integers {1, . . ., n}.Throughout, the symbols n, v, and k are used to represent positive integers.A complete graph on k vertices is denoted K k , and K loop v denotes the graph K v with a loop on each vertex.Definition 1. (Graph-dependent Covering Array) Let G be a loopless directed graph with no parallel arcs, and let V (G) = [k].Let H be a directed graph with no parallel arcs, and let V (H) = [v].A covering array on G and H, denoted CA(n; 2, G, H), is an n × k array with symbols from the alphabet [v] and satisfying the following property: for every arc ij ∈ E(G) and for every arc ab ∈ E(H), there exists a row r l = (r l1 , . . ., r lk ) such that (r li , r lj ) = (a, b).The graphs G and H of a CA(n; 2, G, H) are called the column graph and alphabet graph, respectively.The covering array number for G and H, denoted CAN(2, G, H), is the smallest integer n for which a CA(n; 2, G, H) exists.
In Figure 1, we give two examples of column-alphabet graph pairs with corresponding covering arrays.The left example has undirected column and alphabet graphs, whereas the right example is directed; the underlying graphs of G and H are G and H, respectively.
Let x = (x 1 , . . ., x n ) and y = (y 1 , . . ., y n ) be two v-ary n-tuples, that is, x, y ∈ [v] n .If, for every arc ab ∈ E(H), there exists an index i ∈ [n] such that x i = a and y i = b, then we say that x is H-dependent with y.In a CA(n; 2, G, H), for all arcs ij ∈ E(G), the ith column must be H-dependent with the jth column.
Alphabet graphs and column graphs can in fact be undirected; if so, then our convention is to consider them as directed graphs in which each undirected edge corresponds to two oppositely oriented arcs.With our convention, the notion of H-dependence is a symmetric relation if and only if H is an undirected alphabet graph.If one of G or H is undirected, then, in terms of covering arrays on G and H, we can assume that both G and H are undirected.There may be situations in which it makes sense to allow alphabet graphs and column graphs to have parallel arcs, and to allow column graphs to have loops.Parallel arcs could encode the desired multiplicities of coverage (for specified pairs of symbols indicated by parallel arcs of the alphabet graph or entirely between columns specified by parallel arcs of the column graph).This would be analogous to the "index" parameter of classical covering arrays.In the present paper, we do not consider these situations, but mention them as possible lines of future investigation.Some special cases that have been considered in the literature are the following: • If the alphabet graph is K 2 and the column graph is K k , then the columns of a CA(n; 2, K k , K 2 ) correspond to an antichain of subsets of [n].In this case, Sperner's Theorem [10] gives CAN(2, . • If the alphabet graph is complete on v vertices and has a loop on each vertex and the column graph is complete, then a CA(n; 2, K k , K loop v ) corresponds to a classical v-ary covering array of strength two, denoted CA(n; 2, k, v).The columns of a CA(n; 2, k, v) correspond to what is known as a "qualitatively independent" collection of v-partitions of [n]. 1 For a survey on classical covering arrays, see [1].
• If the alphabet graph is K loop v and the column graph is G, then a CA(n; 2, G, K loop v ) is what is known as a "covering array on G," which we denote by CA(n; 2, G, v).Covering arrays on (undirected column) graphs model testing applications where some factors are known not to interact.See [9] for more details.
• If the alphabet graph is H and the column graph is complete, then the columns of a CA(n; 2, K k , H) correspond to an "H-dependent collection" of |V (H)|-ary n-tuples, that is, a collection C = { x 1 , . . ., x k } in which x i is H-dependent with x j for every ordered pair of two distinct elements x i , x j ∈ C. Asymptotic information-theoretic results related to H-dependent collections have been studied in [2,3].
In Section 2, we define H-dependence graphs and characterize graph-dependent covering arrays in terms of graph homomorphisms to H-dependence graphs.In the remaining sections, we focus on directed column graphs together with the binary alphabet 1 For the particular alphabet graph K loop v , the notion of being K loop v -dependent corresponds to being qualitatively independent.Despite this discord in terminology, we adopted "graph-dependent" since it captures the fact that the symbol coverage of our covering arrays on G and H depends on the edges of both of these accessory graphs.Moreover, the terminology and definition of "graph-dependent partition systems" was already introduced in [3] in order to generalize qualitative independence.
graph T 2 , where V (T 2 ) = {0, 1} and E(T 2 ) = {(0, 1)}.In Section 3, we give bounds for CAN(2, G , T 2 ), where G is a directed column graph.In Section 4, we establish bounds for CAN(2, G , T 2 ), where G is a tournament, that is, an orientation of a complete graph.In this case, determining CAN(2, G , T 2 ) corresponds to an oriented version of Sperner's Theorem.In Section 4.1, we give optimal constructions for covering arrays on transitive tournament column graphs, and we use these in Section 4.2 to obtain tight asymptotic bounds for the covering array number for all tournaments with alphabet graph T 2 .In Section 4.3, we consider another particular family of tournament column graphs which we call circular tournaments.We give constructions for covering arrays on circular tournaments and alphabet graph T 2 ; we show that these constructions are optimal infinitely often.In Section 4.4, we conclude by giving some experimental data and open problems for binary covering arrays on tournaments.

H-dependence graphs
In this section, we give a class of graphs, called H-dependence graphs, which characterizes the problem of determining CAN(2, G, H) in terms of graph homomorphisms.The Hdependence graphs are the natural extension of qualitative independence graphs which were defined by Meagher and Stevens [9] when they characterized covering arrays on (undirected column) graphs via homomorphisms.
Let G and H be graphs.A map f : The next two results are straightforward extensions of analogous results given in [9].Proposition 4. Let G be a column graph and let H be an alphabet graph.Then there exists a CA(n; 2, G, H) if and only if there is a homomorphism G → QI(n, H).In particular, CAN(2, G, H) = min{n : G → QI(n, H)}.
the electronic journal of combinatorics 25(2) (2018), #P2.47 If there exists a CA(n; 2, G, H), then its columns are vertices of QI(n, H).Define a map f : V (G) → V (QI(n, H)) given by f (i) = C i , where C i denotes the ith column of the CA(n; 2, G, H) and , then, for all ab ∈ E(H), there exists an index l ∈ [n] such that c li = a and c lj = b.By definition of the H-dependence graph QI(n, H), the vertex C i is adjacent to C j in QI(n, H).Thus, f defines a homomorphism.
If there exists a homomorphism f : V (G) → V (QI(n, H)), then we can build an n × k array with columns C 1 , . . ., C k given by C i = f (i) for each i ∈ [k].By the definition of QI(n, H), it follows that this array is a CA(n; 2, G, H).Proposition 5. Let H be an alphabet graph and let G 1 and G 2 be column graphs.If there is a homomorphism In particular, for an undirected column graph G, we have where ω(G) and χ(G) denote the clique number and chromatic number of G.
If H 1 and H 2 are alphabet graphs such that V (H 1 ) = V (H 2 ) = [v] and E(H 1 ) ⊆ E(H 2 ), then every pair of H 2 -dependent v-ary n-tuples x and y are necessarily H 1 -dependent.Thus, we have the following lemma.Lemma 6.Let H 1 and H 2 be alphabet graphs such that V (H 1 ) = V (H 2 ) and E(H 1 ) ⊆ E(H 2 ).Then, for every n, the H 2 -dependence graph QI(n, H 2 ) is a subgraph of the H 1dependence graph QI(n, H 1 ).
Going forward, the focus of this paper is on one particular directed alphabet graph: the transitive tournament on two vertices, which we denote by T 2 .The vertex set of T 2 is V (T 2 ) = {0, 1} and its edge set is E(T 2 ) = {(0, 1)}.The alphabet graph T 2 is the natural directed alphabet graph to consider on a binary alphabet.
In Figure 2, we give the T 2 -dependence graphs QI(2, T 2 ) and QI(3, T 2 ).A graph that is not homomorphic to any proper subgraph of itself is called a core.
Since every pair of distinct vertices in the T 2 -dependence graph QI(n, T 2 ) is joined by at least one arc, and since QI(n, T 2 ) has no loops, it follows that QI(n, T 2 ) is a core.
In terms of subsets of [n], a binary n-tuple x is T 2 -dependent with another binary n-tuple y if and only if the corresponding subsets A x , A y ⊆ [n] satisfy A y A x .We often The T 2 -dependence graphs QI(2, T 2 ) and QI(3, T 2 ).make use of this correspondence and refer to vertices of QI(n, T 2 ) as subsets or n-tuples interchangeably.The rank of a binary n-tuple x is the number of times that '1' is an entry of x.Equivalently, the rank of x is the cardinality A and B such that both AB and BA are arcs of QI(n, T 2 ).In particular, as n → ∞, the proportion is the total number of pairs of vertices in QI(n, T 2 ).
Proof.First, we show that the number of pairs of distinct vertices A, B ∈ V (QI(n, T 2 )) such that exactly one of AB and BA is an arc of QI(n, Consider the vertices of QI(n, T 2 ) as subsets of Summing over all sets of rank k and summing over all ranks k ∈ {0, 1, . . ., n}, we have These are the only pairs of vertices for which BA ∈ E(QI(n, T 2 )) and AB ∈ E(QI(n, T 2 )).Thus, there are 3 n − 2 n pairs of distinct vertices in V (QI(n, T 2 )) such that exactly one of AB and BA is an arc of QI(n, T 2 ).
the electronic journal of combinatorics 25(2) (2018), #P2.47 Now, there are 2 n 2 pairs of distinct vertices in QI(n, T 2 ), of which 3 n − 2 n pairs are not symmetrically adjacent.Thus, the number of symmetrically adjacent pairs is 3 Bounds for binary covering arrays on directed column graphs In this section, we investigate the problem of finding CAN(2, G , T 2 ) for directed column graphs G .We give bounds which compare the covering array number for a directed column graph G and alphabet graph T 2 with a binary covering array on the underlying graph G of G .Throughout this section, we use G to denote the underlying graph of a given directed graph G .For a graph G and integers k, v, we denote CAN(2, K k , K loop v ) simply as CAN(2, k, v), and we denote CAN(2, G, K loop v ) by CAN(2, G, v).Theorem 9. Let G be a directed graph with at least one arc, and let G be its underlying graph.Then Proof.For the lower bound, let m = CAN(2, G , T 2 ), and consider a CA(m; 2, G , T 2 ).The columns of this array form a proper vertex-colouring of G since adjacent vertices in G correspond to columns of the CA(m; 2, G , T 2 ) that must be T 2 -dependent in some direction which makes these columns distinct.Therefore χ(G) 2 m , and so log 2 χ(G) m = CAN(2, G , T 2 ).
For the upper bound, let n = CAN(2, G, 2) and consider an optimal CA(n; 2, G, 2).We may assume without loss of generality that the first row is all zeros.Delete this row and we are left with a CA(n − 1; 2, G , K 2 ), for any (directed For G with χ(G) = 2, we have a complete characterization of directed graphs that achieve the lower bound of Theorem 9, based on the following definition and observations.We call a directed graph G a consistently oriented bipartite graph if the underlying graph of G is bipartite and there is a bipartition (X, Y ) of V (G ) such that all arcs of G are directed from X to Y .
If G is a directed graph, then CAN(2, G , T 2 ) can be equivalently defined as the minimum number of consistently oriented bipartite subgraphs into which one can partition the arc set of G .Indeed, the set of arcs of G covered by any particular row of a CA(n; 2, G , T 2 ) forms a consistently oriented bipartite subgraph.In particular, the condition CAN(2, G , T 2 ) = 1 is equivalent to there being a homomorphism G → QI(1, T 2 ) ∼ = T 2 , which is equivalent to G being a consistently oriented bipartite graph.
For χ(G) = 2, the lower bound of Theorem 9 holds with equality if and only if CAN(2, G , T 2 ) = 1, which means the lower bound of Theorem 9 is achieved solely by consistently oriented bipartite graphs.
For higher chromatic numbers, we do not have complete characterizations of graphs achieving the lower bound of Theorem 9; however, for χ(G) = 3, 4, we do have a necessary condition, as follows.If G contains a directed odd cycle, then by [7,Lemma 4.4.11],we have CAN(2, G , T 2 ) 3. For χ(G) = 3, 4, the lower bound of Theorem 9 is given by CAN(2, G , T 2 ) 2. Thus, for these chromatic numbers, a necessary condition to achieve the lower bound of Theorem 9 is for G to be free of any directed odd cycles.
For all chromatic numbers χ 2, there exist directed graphs G for which the lower bound of Theorem 9 holds with equality.Specifically, if G is a transitive ktournament, then the underlying graph of G is K k ; in this case, CAN(2, G , T 2 ) = log 2 k = log 2 χ(G) (see Section 4.1 for more details).
To see that the upper bound of Theorem 9 holds with equality infinitely often, let G be a directed graph such that G contains a directed odd cycle C 2l+1 and the underlying graph More examples of directed column graphs whose covering array numbers achieve the upper bound of Theorem 9 can be found in Appendix A; these examples are tournament column graphs whose underlying graphs have chromatic numbers χ = 3, 5, 9 (see "adjacency vectors" given in Table 2 for k = 3, 5, 9).Tournament column graphs are explored in more detail in Section 4.

Bounds for binary covering arrays on tournaments
In this section, we give bounds on CAN(2, G , T 2 ) where the column graph G is a ktournament.We consider two specific families of tournaments as column graphs: transitive tournaments and "circular" tournaments.We give constructions which are optimal infinitely often for these families of column graphs.
A k-tournament is an orientation of the complete graph K k .Our convention for ktournaments is to label the vertices as 1, 2, . . ., k (except when k = 2 in which case we use 0 and 1).
We are interested in k-tournaments as column graphs with T 2 as the alphabet graph.Among all k-tournaments, is there one with the largest covering array number?Do binary covering arrays on k-tournaments generally require as many rows as classical binary covering arrays?
First, since the T 2 -dependence graph QI(n, T 2 ) has no loops while the underlying graph of a tournament is complete, any homomorphism of a tournament to QI(n, T 2 ) must be vertex-injective.Thus, the analogue of Proposition 4 for binary covering arrays on tournaments is the following.Proposition 10.Let O k be any k-tournament.Then there exists a CA(n; Since the underlying graph of a k-tournament is the complete graph K k with χ(K k ) = k, Theorem 9 gives the following bound.
For k-tournaments, the upper bound of Corollary 11 can be tightened as follows.
Theorem 12. Let k 2 and let O k be any k-tournament.Then .
Proof.The lower bound is given by Corollary 11.To prove the upper bound, we give a construction as follows.Let n be a positive integer such that 2 k 2 n−1 (n−1)/2 .For k-tournament column graphs, the upper bound given in Theorem 12 is an improvement over the upper bound given in Corollary 11 infinitely often.An exact expression for the upper bound of Corollary 11 is known [5,6], and this can be rewritten as follows: For all n, we have 2 n−1 Consequently, for all k 2, we have the electronic journal of combinatorics 25(2) (2018), #P2.47 For each n 1, for all k lying in the range (n−1)/2 , the inequality in (1) is strict.Thus, for infinitely many k-tournaments, the upper bound given in Theorem 12 is a strict improvement over that given in Corollary 11.
For fixed k, we are interested in the spectrum of covering array numbers which arises when we consider all k-tournament column graphs.Are the bounds of Theorem 12 achieved for all k?Among all k-tournaments, which ones have the largest or smallest covering array numbers?
For the lower bound of Theorem 12, we have a complete answer by applying Proposition 10.
Theorem 13.A k-tournament O k achieves the lower bound of Theorem 12 if and only if O k is a subgraph of the T 2 -dependence graph QI( log 2 k , T 2 ).

Transitive tournaments
Aside from the general answer given by Theorem 13, we also have a specific infinite family of tournament column graphs achieving the lower bound of Theorem 12, namely, transitive tournaments.
Let k 2. The transitive k-tournament, denoted T k , is the tournament with V (T k ) = [k] and arcs ij ∈ E(T k ) if and only if i < j.
Proof.Let A and B be subsets of [n] (corresponding to vertices of QI(n, T 2 )).Notice that AB ∈ E(QI(n, T 2 )) if and only if B A. We can thus order the vertices of QI(n, T 2 ) in non-descending order of rank, and this ordering has the property that AB ∈ E(QI(n, T 2 )) whenever A precedes B. With this ordering in place, it is clear that QI(n, T 2 ) contains T 2 n as an induced subgraph since there is a strict total ordering on the vertices of T 2 n .It now follows that for all k 2 n there exist homomorphisms T k → T 2 n → QI(n, T 2 ).By Proposition 4, we have CAN(2, T k , T 2 ) log 2 k .By Theorem 12, we have CAN(2, T k , T 2 ) log 2 k , which completes the proof.
The proof of Theorem 14 provides a simple construction for CA( log 2 k ; 2, T k , T 2 ).If n = log 2 k , then k columns corresponding to subsets of [n] in non-descending order of rank form the columns of a CA( log 2 k ; 2, T k , T 2 ).In the following theorem, we use this construction for transitive tournaments to build binary covering arrays.The construction we provide in Theorem 15 is not optimal in general; however, it is optimal asymptotically, and it produces binary covering arrays built from blocks of rows corresponding to the simple construction for binary covering arrays on transitive tournaments.
the electronic journal of combinatorics 25(2) (2018), #P2.47 Proof.Let n = log 2 k , and let m be the minimum number of binomial coefficients needed to write k Now, build a CA(n; 2, T k , T 2 ) as described in the proof of Theorem 14.Without loss of generality, we may assume that the columns of the CA(n; 2, T k , T 2 ) correspond to the subsets of [n] of ranks a 1 , a 2 , . . ., a m−1 and (as many as needed of) the a m -subsets of [n], sorted in non-decreasing order of rank.Note, for each i ∈ [m], the columns corresponding to the a i -subsets form an antichain and are thus already K 2 -dependent.We extend this CA(n; 2, T k , T 2 ) into a CA(n + log 2 m ; 2, K k , K 2 ) by appending to it at most log 2 ( log 2 k + 1) additional rows.These additional rows correspond to the rows of a CA( log 2 m ; 2, T m , T 2 ) whose columns we denote by C 1 , C 2 , . . ., C m .Under each column of the CA(n; 2, T k , T 2 ) that corresponds to an a i -subset of [n], we put a copy of C m−i+1 , as depicted in Figure 3.The array given in Figure 3 is indeed a CA(n + log 2 m ; 2, K k , K 2 ).Thus, we have By adding two constant rows to the CA(n + log 2 m ; 2, K k , K 2 ), namely a row of all zeros and another row of all ones, we get a CA(n + log 2 m + 2; 2, k, 2).

Asymptotic bounds
For directed column graphs G and alphabet graph T 2 , we now show that, asymptotically, CAN(2, G , T 2 ) grows logarithmically with respect to the chromatic number of the underlying graph of G .
Theorem 16.For each c 2, let G c be some directed graph with underlying graph G c satisfying χ(G c ) = c.Then the electronic journal of combinatorics 25(2) (2018), #P2.47 Proof.Since CAN(2, G c , T 2 ) CAN(2, χ(G c ), 2), it follows from Theorems 9 and 15 that, for each c 2, we have Clearly, the above bounds are asymptotically equal to log 2 χ(G c ), as c → ∞.
Corollary 17.For each k 2, let O k denote some k-tournament.Then

Circular tournaments
In addition to transitive tournaments, we consider one other infinite family of tournament column graphs, which we call circular tournaments.We prove that the covering array number for circular k-tournaments and alphabet graph T 2 is always log 2 k or log 2 k +1.
In Figure 4, we depict several circular k-tournaments.Determining CAN(2, Ω k , T 2 ) for all k is an interesting extremal problem that corresponds to an oriented version of Sperner's Theorem [10].In terms of covering arrays, Sperner's Theorem determines CAN(2, K k , K 2 ) for all k.
Here, we give some important properties of circular tournaments.
Proposition 19.If k is odd, then the circular tournament Ω k is vertex-transitive.If k is even, then the automorphism group of Ω k is trivial.
Proof.Let k be odd, and let i, j ∈ V (Ω k ) be two distinct vertices.We can write j = i + x (mod k) for a unique x ∈ {1, . . ., k − 1}.The map f : given by the electronic journal of combinatorics 25(2) (2018), #P2.47 f (u) = u + x (mod k) is an automorphism such that f (i) = j.Thus, when k is odd, Ω k is vertex-transitive, as claimed.Now, let k be even and write k = 2l.The vertices of Ω k are of two types: the vertices labelled 1, 2, . . ., l which have outdegree l and indegree l − 1, and the vertices labelled l+1, l+2, . . ., 2l which have outdegree l−1 and indegree l.Suppose f : V (Ω k ) → V (Ω k ) is an automorphism such that f (i) = j for some vertices i, j ∈ V (Ω k ).If i ∈ {1, . . ., l}, then in order for its in-and outdegree to match, j must also be from among the vertices in the set {1, . . ., l}.In order for the neighbours of i and j to have the correct in-and outdegrees, both i and j must have the same number of neighbours from the set {l + 1, . . ., 2l}.This can happen only if i = j.The argument when i ∈ {l + 1, . . ., 2l} is similar.Thus, when k is even, the automorphism group of Ω k is trivial.
Proposition 20.For all k 3, the circular tournament Ω k contains an induced subgraph isomorphic to Ω k−1 .In particular, when k = 2l + 1, we can delete any vertex to obtain a copy of Ω 2l .When k = 2l, we can delete either vertex l or l + 1 from Ω 2l in order to obtain a copy of Ω 2l−1 .
By Proposition 19, Ω 2l+1 is vertex-transitive.Therefore, we can delete any vertex of Ω 2l+1 in order to obtain a copy of Ω 2l .Now, let k = 2l and let X denote Ω k with the vertex l deleted.We claim that . Consider the following cases: By definition of Ω 2l−1 , we must have j = i + t for some t ∈ {1, . . ., l − 2}.Thus, j = i + t for some t < l, and by definition of Ω 2l , we have f (i)f (j) ∈ E(X).
Since the above cases cover all arcs of the tournament Ω 2l−1 , it follows that f is an isomorphism.Similarly, we can show that Ω 2l−1 is isomorphic to the subgraph of Ω 2l in which the vertex l − 1 is deleted.
Next, we give a recursive construction for covering arrays with alphabet graph T 2 in which we make use of a supergraph of the circular tournament, which we denote Ω + k , defined for even values of k as follows.Let k be even.The vertex set of and only if j = i + t for some t such that 1 t k/2 (addition is done modulo k).The difference between Ω k and Ω + k is that Ω + k has a pair of oppositely oriented arcs joining the vertices i, j ∈ [k] with |j − i| = k/2.In particular, Ω + k is not a tournament, but does contain subgraphs isomorphic to Ω m , for all m k.Consequently, we have CAN(2, Ω m , T 2 ) CAN(2, Ω + k , T 2 ) for all circular tournaments Ω m with m k.
Proposition 21.Let k be even.Then CAN(2, . Take two copies of this array and interleave their columns as shown in Figure 5. Add a row of alternating zeros and ones.Let us call the array we obtain A.
Figure 5: Recursive construction for a graph-dependent covering array on Ω + 2k and T 2 .
For each j ∈ [2k], the jth column of A corresponds to C j/2 (the j/2 th column of the CA(n; 2, Ω + k , T 2 )).Let i and j be the indices of two distinct columns of A such that j = i + t (mod 2k) for some t such that 1 t k.We must show that some row of A has a zero in the ith column and a one in the jth column; if there exists such a row, then for short, we say that A covers {(i, 0), (j, 1)}.

Data and questions for binary covering arrays on tournaments
Transitive tournaments and circular tournaments meet the lower bound of Theorem 12 for infinitely many values of k.We are interested in knowing how often the upper bound of Theorem 12 is met, or whether there is a tighter bound for tournament column graphs.
the electronic journal of combinatorics 25(2) (2018), #P2.47 Using a computer search and the database of tournaments given in [8], for each k 9, we determined CAN(2, O k , T 2 ) for every k-tournament O k (up to isomorphism).Our findings are summarized in Table 1 For circular tournaments in particular, Theorem 22 determines CAN(2, Ω k , T 2 ) to within 1 row.The following problem is also of interest.It was conjectured in [7,Conjecture 4.5.1] that circular tournaments constitute an extremal family of tournaments as described in Problem 29; however, our data for 9tournaments shows that this is not the case.Adjacency matrices and other properties for the two exceptional 9-tournaments referred to in Table 1 are given in Appendix A.1.

A Extremal k-tournaments for 2 k 9
Up to k = 9, we determined CAN(2, O k , T 2 ) for all pairwise non-isomorphic k-tournaments O k by using a computer search and the lists of all k-tournaments given in [8].For each k-tournament O k with k 9, we found that CAN(2, O k , T 2 ) ∈ { log 2 k , log 2 k + 1}.In the following table, for each k such that 2 k 9, we give the lower and upper bounds of Theorem 12, denoted L.B. and U.B., respectively.We write CAN to abbreviate the covering array number for a covering array on a given tournament column graph with alphabet graph T 2 .For 2 k 9, we give the number of k-tournaments with CAN = n for each possible value of n in the range L.B. n U.B..For some parameters, the actual tournaments are given as the upper triangle of the adjacency matrix in row order, on one line without spaces.We refer to this representation as the adjacency vector.For example, the 4-tournament represented by the adjacency vector 000111 has the following adjacency matrix and graph: By permuting the rows and columns of its adjacency matrix, any given tournament has several distinct adjacency vector representations.In Table 2, each given adjacency vector is represented exactly as given in the lists in [8], with the exception of the 9tournaments; the adjacency vectors of the given 9-tournaments have been reconfigured in order to emphasize their structure which we discuss in more detail in Appendix A.1.

1 )/ 2 . 2 - 2 - 2 - 2 -
Take any maximum matching M of K k (the underlying graph of O k ).Partition V (O k ) into two parts of sizes k/2 and k/2 such that the ends of all edges in M are oriented from the first part to the second in O k (if k is odd, we add the unmatched vertex to the first part).Relabel the vertices of O k so that the vertices of the first part of the partition are labelled 1, . . ., k/2 , and the vertices of the second part are labelled k/2 + 1, . . ., k.To construct a CA(n; 2, O k , T 2 ), we first build an (n − 1) × k array with columns C 1 , . . ., C k corresponding to n−1 subsets of [n − 1] as follows.Take k/2 of the n−1 subsets of [n − 1] in some order, followed by the same first k/2 columns repeated in the same order, that is, C 1 , . . ., C k/2 are distinct n−1 sets and C i+ k/2 = C i for 1 i k/2 .Add an nth row to cover the arcs of M .This additional row has zeros as its first k/2 entries and ones as its last k/2 entries.Since every two distinct n−1 sets are K 2 -dependent, any two distinct columns C i and C j (of length n − 1) are K 2 -dependent except when |j − i| = k/2 .The extra row ensures that we cover the arcs of M .Note, when k = 2, this extra row is in fact the only row of the constructed array.Thus, we have constructed a CA(n; 2, O k , T 2 ), and it follows that CAN(2, O k , T 2 ) n whenever k 2 n−1

Problem 28 .
For each L 2, for all k in the range 2 L−1 < k 2 L , the covering array number CAN(2, Ω k , T 2 ) ranges from log 2 k to log 2 k + 1, non-decreasingly as k increases.For all L 2, for each range 2L−1 < k 2 L , determine the threshold value of k for which CAN(2, Ω k , T 2 ) = log 2 k and CAN(2, Ω k+1 , T 2 ) = log 2 k + 1.Aside from a tight numerical upper bound on CAN(2, O k , T 2 ), we are interested in finding an extremal family of k-tournaments.In particular, we wish to have a structural characterization of some infinite family of k-tournaments, say{X k } ∞ k=2 , for which CAN(2, O k , T 2 ) CAN(2, X k , T 2 ) for every k-tournament O k .theelectronic journal of combinatorics 25(2) (2018), #P2.47 Problem 29.Find and characterize an infinite family {X k } ∞ k=2 of k-tournaments for which CAN(2, O k , T 2 ) CAN(2, X k , T 2 ) for every k-tournament O k .

Table 1 :
and more details are given in Appendix A. In Table1, we use L.B. and U.B. to denote the lower and upper bounds of Theorem 12, respectively.In Table1, we write CAN to abbreviate the covering array number on a given tournament column graph with alphabet graph T 2 .Summary of analysis for small tournaments.Our analysis of small tournaments shows that for k ∈ {2, 3, 4, 5, 6, 9}, the upper bound of Theorem 12 is attained.Determine whether the upper bound of Theorem 12 is met infinitely often.If not, determine a tight numerical upper bound on CAN(2, O k , T 2 ) for any k-tournament O k .Based on our data in Table1, we ask whether the bounds of Theorem 22 hold for all k-tournaments.

Table 2 :
Data for small tournaments.