Some Covering and Packing Problems for Mixed Triples

Abstract

A mixed graph has both edges and directed edges (or “arcs”). A complete mixed graph on v vertices, denoted $M_v$, has, for every pair of vertices u and v, an edge $\{u,v\}$, an arc $(u,v)$, and an arc $(v,u)$. A decomposition of the complete mixed graph on v vertices into a partial orientation of a three-cycle with one edge and two arcs (of which there are three types) is a mixed triple system of order v. Necessary and sufficient conditions for the existence of a mixed triple system of order v are well known. In this work packings and coverings of the complete mixed graph with mixed triples are considered. Necessary conditions are given for each of the three relevant mixed triples, and these conditions are shown to be sufficient for two of the relevant mixed triples. For the third mixed triple, a conjecture is given concerning the sufficient conditions. Applications of triple systems in general are discussed, as well as possible applications of mixed graphs, mixed triple systems, and packings and coverings with mixed triples.

1. Introduction

1.1. Outline

In this paper, we consider a covering and a packing problem in the setting of mixed graphs. In Section 1.2 we give basic definitions concerning graph and digraph decompositions and illustrate them with triple systems: Steiner triple systems, Mendelsohn triple systems, and directed triple systems. Section 1.3 introduces graph and digraph coverings and packings and explains why they are desirable structures to explore when corresponding decompositions do not exist. Section 1.4 addresses a general application of Steiner triple systems and an agriculture-motivated application of Mendelsohn and directed triple systems (namely, crop rotation). In Section 1.5, we define a mixed graph and a mixed graph decomposition and illustrate this with mixed triple systems. Section 1.6 gives another agricultural application of mixed graphs and mixed triple systems. After introducing the idea of companion planting along with crop rotation, experimental designs are described using these structures. In Section 2, we lay out the equipment needed for the proofs of our main results. In particular, in the complete mixed graph, “edge difference” and “arc difference” are defined, and their use in creating decompositions is described. We will use difference methods in mixed graph covering and packing constructions. Section 3 includes necessary and sufficient conditions for certain mixed graph packings, and Section 4 includes necessary and sufficient conditions for certain mixed graph coverings. A solution of these problems in all relevant cases is not given, but a conjecture is offered for the case that is left unsolved. In Section 5, we discuss future research in this area and other possible applications of mixed graphs.

1.2. Graph and Digraph Decompositions

We denote the edge set of a graph G as $E\left(G\right)$, and the arc set of a digraph D as $A\left(D\right)$. We use the same notation for mixed graphs in Section 3 and Section 4. We start with the definition of certain graph and digraph decompositions and examples of each. The definitions given in Section 1.2 and Section 1.3 can be found in [1] (though in somewhat different notation than that used here).

Definition 1.

An isomorphic γ-decomposition of the complete graph $K_v$ is a set of subgraphs of $K_v$, $\{{\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n\}$, such that each ${\gamma }_i$ is isomorphic to γ, the ${\gamma }_i$s are edge disjoint (that is, $E\left({\gamma }_i\right)\cap E\left({\gamma }_j\right)=⌀$ for $i\ne j$), and ${\cup }_{i=1}^{n}E\left({\gamma }_i\right)=E\left(K_v\right)$. An isomorphic δ-decomposition of the complete digraph $D_v$ is a set of subdigraphs of $D_v$, $\{{\delta }_1,{\delta }_2,\dots ,{\delta }_n\}$, such that each ${\delta }_i$ is isomorphic to δ, the ${\delta }_i$s are arc disjoint (that is, $A\left({\delta }_i\right)\cap A\left({\delta }_j\right)=⌀$ for $i\ne j$), and ${\cup }_{i=1}^{n}A\left({\delta }_i\right)=A\left(D_v\right)$.

A Steiner triple system of order v is an isomorphic three-cycle decomposition of $K_v$. Historically, Steiner triple systems were the first graph decompositions explored, and it is well known that a Steiner triple system of order v exists if and only if $v\equiv 1$ or 3 (mod 6). For references and a detailed history, see [1].

The digraph decompositions relevant to this work are decompositions of $D_v$ into one of the two orientations of a three-cycle. The two orientations are the directed triple and the Mendelsohn triple, as given in Figure 1. A decomposition of $D_v$ into directed triples is a directed triple system of order v, and a decomposition of $D_v$ into Mendelsohn triples is a Mendelsohn triple system of order v. Each such decomposition exists if and only if $v\equiv 0$ or 1 (mod 3), except in the case of a Mendelsohn triple system of order 6, which does not exist [2,3].

1.3. Graph and Digraph Coverings and Packings

When a Steiner triple system of order v does not exist, for example, for $v\equiv 2$ (mod 6), we can explore approximations of Steiner triple systems. There are two approaches. We can insist that every edge of $K_v$ be contained in some three-cycle in the set of three-cycle subgraphs of $K_v$ and drop the condition of edge disjointness among the three-cycles in the set. Alternatively, we can insist that the disjointness among the three-cycles hold, but drop the condition that every edge of $K_v$ is in some three-cycle. These approaches lead to the ideas of a covering and a packing, respectively, of $K_v$. More formally, we have the following.

Definition 2.

An isomorphic γ-covering of the complete graph $K_v$ is a set of subgraphs of $K_v$, $\{{\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n\}$, such that each ${\gamma }_i$ is isomorphic to γ, and ${\cup }_{i=1}^{n}E\left({\gamma }_i\right)=E\left(K_v\right)$. A minimal isomorphic γ-covering of $K_v$ is an isomorphic γ-covering of $K_v$ such that the total number of edges in the covering, ${\sum }_{i=1}^n\left|E\left({\gamma }_i\right)\right|$, is minimal. An isomorphic γ-packing of the complete graph $K_v$ is a set of subgraphs of $K_v$, $\{{\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n\}$, such that each ${\gamma }_i$ is isomorphic to γ, and the ${\gamma }_i$s are edge disjoint (i.e., $E\left({\gamma }_i\right)\cap E\left({\gamma }_j\right)=⌀$ for $i\ne j$). A maximal isomorphic γ-packing of $K_v$ is an isomorphic γ-packing of $K_v$ such that the total number of edges in the packing, ${\sum }_{i=1}^n\left|E\left({\gamma }_i\right)\right|$, is maximal.

Necessary and sufficient conditions for minimal coverings and maximal packings of $K_v$ with three-cycles are known [4,5,6]. One might notice the similarity between a minimal graph covering of $K_v$ and the idea of the outer measure of a set of real numbers, as well as the similarity between a maximal graph packing of $K_v$ and the idea of the inner measure of a set of real numbers. In the real setting, the outer measure is defined in terms of an infimum, and the inner measure is defined in terms of a supremum [7], but this is replaced with a minimum and maximum in our discrete case. Coverings and packings are similarly defined for digraphs.

Definition 3.

An isomorphic δ-covering of the complete digraph $D_v$ is a set of subdigraphs of $D_v$, $\{{\delta }_1,{\delta }_2,\dots ,{\delta }_n\}$, such that each ${\delta }_i$ is isomorphic to δ, and ${\cup }_{i=1}^{n}A\left({\delta }_i\right)=A\left(D_v\right)$. A minimal isomorphic δ-covering of $D_v$ is an isomorphic δ-covering of $D_v$ such that the total number of arcs in the covering, ${\sum }_{i=1}^n\left|A\left({\gamma }_i\right)\right|$, is minimal. An isomorphic γ-packing of the complete digraph $D_v$ is a set of subdigraphs of $D_v$, $\{{\delta }_1,{\delta }_2,\dots ,{\delta }_n\}$, such that each ${\delta }_i$ is isomorphic to δ, and the ${\delta }_i$s are arc disjoint (i.e., $A\left({\delta }_i\right)\cap A\left({\delta }_j\right)=⌀$ for $i\ne j$). A maximal isomorphic δ-packing of $D_v$ is an isomorphic δ-packing of $D_v$ such that the total number of arcs in the packing, ${\sum }_{i=1}^n\left|A\left({\delta }_i\right)\right|$, is maximal.

Necessary and sufficient conditions for coverings and packings of $D_v$ with directed triples and with Mendelsohn triples are given in [8].

1.4. Applications Related to Experimental Design

Being from the area of design theory, Steiner triple systems have applications in various areas of experimental design and the partitioning of samples for measurement. Consider, for example, a machine that compares three samples at a time. Three samples are entered into the machine and compared pairwise. However, two samples can only be compared by being run together in the machine (say that it cannot be calibrated between runs), and the machine must run samples three at a time. If a collection of v samples need to be compared pairwise, how can this be realized efficiently? If $v\equiv 1$ or 3 (mod 6), then this can be realized, and the solution as to how to run the samples through the machine is given by a Steiner triple system of order v. Each sample is represented by a vertex of the complete graph $K_v$, an edge joining two vertices represents a comparison of the two corresponding samples, and a three-cycle (i.e., a triple) of the Steiner triple system represents a run of the machine. However, if v is not 1 or 3 (mod 6), then there is no perfect solution as to how the samples should be processed. This leads to two approaches. If we insist on comparing all samples, then some samples will have to be compared to each other more than once. An optimal approach to this would require a minimal number of repetitions. A minimal three-cycle covering of $K_v$ would yield one of the best possible solutions in this case. On the other hand, it might be expensive to run the machine, and we would want to compare as many pairs as possible, without repetition. A three-cycle packing of $K_v$ would yield one of the best possible solutions in this case.

Crop rotation is a common agricultural practice. It involves planting different crops (sequentially) in a field so that one crop is planted one year and a different crop is planted the next year. The United States Department of Agriculture gives suggestions for and benefits of crop rotation as follows [9]:

“For crop rotation to be most effective, don’t plant an area with vegetables from the same plant family more than once every three to four years. Rotating crops can have important production benefits such as increasing yields, improving nutrients and organic matter in the soil, and it can help disrupt the lifecycle of crop pests, reducing chemical use.”

If we represent the crops in a proposed plan of crop rotation as vertices, and let an arc from vertex a to vertex b represent the situation where crop a is planted in the field the year before crop b, then we can store all possible sequences of planting the crops one before the other as a complete digraph. If we were to carry out a measurement of some property associated with each arc (such as crop yield) using a machine similar to the one described above for an application of Steiner triple systems (which can only run three samples at a time), then measuring all such properties of arcs could be accomplished using a Mendelsohn triple system or a directed triple system. In fact, if we are only concerned with measuring three properties of arcs at a time, then we could use an oriented triple system; this is a decomposition of the complete digraph into orientations of a three-cycle where some of the oriented three-cycles are Mendelsohn triples and the remainder are directed triples. Oriented triple systems (also called “ordered triple systems”) exist if and only if the order is $v\equiv 0$ or 1 (mod 3) [10]. The desirability of corresponding packings and coverings apply here when a decomposition does not exist.

1.5. Mixed Graphs

We define a mixed graph on v vertices in terms of the vertex set and the set of arcs and edges. A mixed graph $(V,C)$ has vertex set V, where $\left|V\right|=v$, and arc/edge set C, where the elements of C are ordered pairs (called arcs) and unordered pairs (called edges) of elements of V. We denote the set of edges of $(V,C)$ as E and the set of arcs as A, so that $C=E\cup A$. As with graphs and digraphs, the complete mixed graph on v vertices, denoted $M_v$, is the mixed graph $(V,C)$, where for every pair of distinct vertices x and y, set C contains an edge joining x and y, an arc joining x to y, and an arc joining y to x. The following definition appears in [11].

Definition 4.

For a mixed graph μ, a μ-decomposition of $M_v$ is a set, $\{{\mu }_1,{\mu }_2,\dots ,{\mu }_n\}$, of edge disjoint and arc disjoint isomorphic copies of μ such that the union of the edge sets and arc sets of the ${\mu }_i$ are the edge set and the arc set, respectively, of the complete mixed graph:

$${\cup }_{i=1}^{n}E\left({\mu }_i\right)=E\left(M_v\right)and{\cup }_{i=1}^{n}A\left({\mu }_i\right)=A\left(M_v\right).$$

The complete mixed graph has twice as many arcs as edges, so if we consider a $\mu$-decomposition of $M_v$, then mixed graph $\mu$ must have twice as many arcs as edges. Our interest lies with $\mu$ as a mixed graph with an underlying graph as a three-cycle. There are three partial orientations of a three-cycle with one edge and two arcs. These partial orientations, which we call mixed triples, are given in Figure 2.

We follow the notation given in Figure 2 and call a $T_i$-decomposition of $M_v$ a $T_i$-triple system for $i\in \{1,2,3\}$. The existence of mixed triple systems of order v are classified in [11].

Theorem 1

([11]). A decomposition of the complete mixed graph of order v, $M_v$, exists for $T_1$, $T_2$, and $T_3$ if and only if $v\equiv 1$ (mod $2)$, except for $v\in \{3,5\}$ for the case of $T_3$.

Just as there are graph and digraph packings and coverings of complete graphs and digraphs, we can define mixed graph packings and coverings of the complete mixed graph.

Definition 5.

An isomorphic μ-covering of the complete mixed graph $M_v$ is a set of mixed subgraphs of $M_v$, $\{{\mu }_1,{\mu }_2,\dots ,{\mu }_n\}$, such that each ${\mu }_i$ is isomorphic to μ, ${\cup }_{i=1}^{n}E\left({\mu }_i\right)=E\left(M_v\right)$, and ${\cup }_{i=1}^{n}A\left({\mu }_i\right)=A\left(M_v\right)$. The padding P of the covering is the multiset of edges and arcs of $M_v$ that are included in the covering more than once (repeated according to multiplicity). A minimal isomorphic μ-covering of $M_v$ is an isomorphic μ-covering of $M_v$ such that the sum of the number of edges and number of arcs in the covering, ${\sum }_{i=1}^n(|E\left({\mu }_i\right)|+|A\left({\mu }_i\right)|)$, is minimal. An isomorphic μ-packing of the complete mixed graph $M_v$, is a set of mixed subgraphs of $M_v$, $\{{\mu }_1,{\mu }_2,\dots ,{\mu }_n\}$, such that each ${\mu }_i$ is isomorphic to μ, and the ${\mu }_i$s are edge and arc disjoint (i.e., $E\left({\mu }_i\right)\cap E\left({\mu }_j\right)=⌀$ and $A\left({\mu }_i\right)\cap A\left({\mu }_j\right)=⌀$ for $i\ne j$). The leave L of the packing is the set of edges and arcs of $M_v$ that do not appear in the packing. A maximal isomorphic μ-packing of $M_v$ is an isomorphic μ-packing of $M_v$ such that the total number of edges and arcs in the packing, ${\sum }_{i=1}^n(|E\left({\mu }_i\right)|+|A\left({\mu }_i\right)|)$, is maximal.

The purpose of this paper is to explore maximal packings and minimal coverings of $M_v$ with the mixed graphs $T_1$, $T_2$, and $T_3$.

1.6. An Application of Mixed Graphs

In terms of applications, we consider the example of planting crops again. As discussed above, crop rotation in which one crop is planted in a field one year followed by a different crop the next year can be modeled using digraphs. Another common agricultural practice is companion planting. Companion planting involves planting two crops together in a field. According to the West Virginia University Extension Service [12], “Using different types of plants [together] can help deter harmful insects, provide support for crops, offer shade to smaller plants, provide weed suppression, attract beneficial insects, as well as increase your overall soil health”. If we were to combine crop rotation and companion planting, then the sequential planting of crop rotation can be represented by arcs, and the companion planting (in which two crops are planted together in the same year) can be represented by edges. The combination is then represented by a mixed graph. Again, if we measure some property associated with each edge and with each arc (such as crop yield) with a machine that can only handle three samples at a time, then measuring all samples could be accomplished using a mixed triple system. As mentioned in the situation for crop rotation alone (and the use of digraphs), we could also use a “hybrid mixed triple system” in which some of the triples are of type $T_1$, some are of type $T_2$, and the remainder are of type $T_3$ (we might even allow other partial orientations of the three-cycle, but such structures have not yet been studied). Again, the desirability of corresponding packings and coverings apply here when a decomposition does not exist.

2. More Mixed Graph Definitions and Difference Methods

We denote an edge joining vertices x and y, where we require $x\ne y$, of a mixed graph as $\{x,y\}$. We denote an arc from vertex x to vertex y, where again $x\ne y$, as the ordered pair $(x,y)$. For arc $(x,y)$, vertex x is called the tail, and vertex y is called the head. So, the mixed triple $T_1={[a,b,c]}_1$, for example, consists of edge $\{b,c\}$ and arcs $(a,b)$ and $(a,c)$. The converse of a given mixed graph is obtained by reversing all the arcs of the graph. More precisely, the converse of mixed graph M is the mixed graph $M^{\prime }$ such that $V\left(M^{\prime }\right)=V\left(M\right)$, $E\left(M^{\prime }\right)=E\left(M\right)$, and $A\left(M^{\prime }\right)=\{(y,x)\mid (x,y)\in A\left(M\right)\}$. For example, the converse of $T_1$ is $T_2$, and $T_3$ is self-converse (that is, $T_3$ is isomorphic to its converse).

Definition 6.

For a vertex v of a mixed graph, we define the edge degree of v, denoted $d_e\left(v\right)$, as the number of edges that have v as an end. The in-degree of v, denoted $d_i\left(v\right)$, is the number of arcs that have v as its head. The out-degree of v, denoted $d_o\left(v\right)$, is the number of arcs that have v as its tail. The total degree of v, denoted $d\left(v\right)$, is $d\left(v\right)=d_e\left(v\right)+d_i\left(v\right)+d_o\left(v\right)$.

In mixed graph $T_1={[a,b,c]}_1$, the various degrees satisfy $d_e\left(a\right)=d_i\left(a\right)=d_o\left(b\right)=d_o\left(c\right)=0$, $d_e\left(b\right)=d_i\left(b\right)=d_e\left(c\right)=d_i\left(c\right)=1$, and $d_o\left(a\right)=d\left(a\right)=d\left(b\right)=d\left(c\right)=2$. The following definition and discussion appear in [13]. They are given here to clarify the proof technique.

Definition 7.

We take the vertex set of $M_v$ to be $V=\{0,1,2,\dots ,v-1\}$. Then, the edge difference associated with edge $\{x,y\}$ is defined as $min\left\{\right(x-y\left)\right(modv),(y-x\left)\right(modv\left)\right\}$. The arc difference associated with arc $(x,y)$ is $(y-x)\left(modv\right)$. The set of edge differences for $M_v$ is then $\{1,2,\dots ,\lfloor v/2\rfloor \}$, and the set of arc differences is $\{1,2,\dots ,v-1\}$.

We now describe a common technique of proof known as difference methods. With the notation of Definition 7, consider the permutation $\pi =(0,1,2,\dots ,v-1)$ of vertex set V. The orbit of an edge $\{x,y\}$ under $\pi$ includes all edges of $M_v$, with the same edge difference as edge $\{x,y\}$ (namely, edge difference $min\left\{\right(x-y\left)\right(\mathrm{mod}v),(y-x\left)\right(\mathrm{mod}v\left)\right\}$). The orbit of arc $(x,y)$ under $\pi$ includes all arcs of $M_v$ with the same arc difference as arc $(x,y)$ (namely, arc difference $(y-x)\left(\mathrm{mod}v\right)$). Consider a set of subgraphs (or “sub-mixed-graphs”) of $M_v$, say $\{b_1,b_2,\dots ,b_k\}$, where each $b_i\cong \mu$, each edge difference $1,2,\dots ,\lfloor v/2\rfloor$ is associated with an edge of exactly one of the $b_i$, and each arc difference $1,2,\dots ,v-1$ is associated with an arc of exactly one of the $b_i$. The images of all of the $b_i$ under the power of $\pi$ produce a collection of isomorphic copies of $\mu$, $\{{\pi }^i\left(b_1\right),{\pi }^i\left(b_2\right),\dots ,{\pi }^i\left(b_k\right)\mid i=0,1,\dots ,v-1\}$, such that every edge and every arc of $M_v$ occur exactly once in the set of subgraphs. That is, the set forms a $\mu$-decomposition of $M_v$. Such a set is called a set of base blocks for the decomposition.

3. Packing the Complete Mixed Graph with Mixed Triples

We now present necessary conditions for the existence of a maximal packing of the complete mixed graph with each of the mixed triples, $T_1$, $T_2$, and $T_3$. We prove that the necessary conditions are in fact sufficient for $T_1$ and $T_2$. We do not address sufficiency for maximal $T_3$-packings, but we conjecture that the necessary conditions we give for these are also sufficient.

Lemma 1.

In a packing of $M_v$ with $T_1$ where v is even, the leave consists of at least $v/2$ edges and at least v arcs. The same holds for packings of $M_v$ with $T_2$.

Proof. 

For even v, each vertex of $M_v$ is of out-degree $v-1$, which is odd. Since in $T_1$, each vertex is of an even out-degree, in any packing of $M_v$ with $T_1$, the leave is some mixed graph in which each vertex is of an odd out-degree. That is, each vertex of the leave is of an out-degree of at least one. So, the leave must contain at least v arcs. Since both $M_v$ and $T_1$ contain twice as many arcs as edges, this must be a property of the leave of a packing as well. Hence, the leave must contain at least $v/2$ edges and at least v arcs. The same argument holds for $T_2$, where “out-degree” is replaced with “in-degree”. □

In Theorem 2 below, we show that the lower bounds given in Lemma 1 are, in fact, attained in a maximal packing.

Lemma 2.

In a packing of $M_v$ with $T_3$ where v is even, the leave consists of the following:

1. 

At least $v/6$ edges and at least $v/3$ arcs when $v\equiv 0$ (mod $6)$;

2. 

At least $(v+4)/6$ edges and at least $(v+4)/3$ arcs when $v\equiv 2$ (mod $6)$;

3. 

At least $(v+2)/6$ edges and at least $(v+2)/3$ arcs when $v\equiv 4$ (mod $6)$.

Proof. 

For even v, each vertex of $M_v$ is of a total degree $3(v-1)$, which is odd. Since in $T_3$, each vertex is of an even total degree, in any packing of $M_v$ with $T_3$, the leave is some mixed graph in which each vertex is of an odd total degree. That is, each vertex of the leave is of a total degree at least one. So, there must be at least $v/2$ edges and arcs total in the leave. Since both $M_v$ and $T_3$ contain twice as many arcs as edges, this must be a property of the leave of a packing as well. So, the leave must have a total number of edges and arcs that is a multiple of three. That is, the total number of edges is at least $\lceil v/6\rceil$, and the total number of arcs in the leave is at least $2\lceil v/6\rceil$. For $v\equiv 0$ (mod 6), we have $\lceil v/6\rceil =v/6$ and $2\lceil v/6\rceil =v/3$. For $v\equiv 2$ (mod 6), we have $\lceil v/6\rceil =(v+4)/6$ and $2\lceil v/6\rceil =(v+4)/3$. For $v\equiv 4$ (mod 6), we have $\lceil v/6\rceil =(v+2)/6$ and $2\lceil v/6\rceil =(v+2)/3$. So, the claimed lower bounds are established. □

Theorem 2.

In a maximal packing of $M_v$ with $T_1$, where $v\ge 3$, the leave L satisfies the following:

1. 

$E\left(L\right)=A\left(L\right)=⌀$ for v odd;

2. 

$\left|E\right(L\left)\right|=v/2$ and $\left|A\right(L\left)\right|=v$ for v even.

The same holds for maximal packings of $M_v$ with $T_2$.

Proof. 

If v is odd, then a $T_1$-decomposition and a $T_2$-decomposition of $M_v$ exist by Theorem 1 so that $E\left(L\right)=A\left(L\right)=⌀$ in this case, as claimed.

Case 1. Suppose $v\equiv 0$ (mod 4), say with $v=4k$ where $k\ge 1$. Consider the collection of $T_1$s:

$$\{{[i,k-r+i,k+r+1+i]}_1\mid r=0,1,\dots ,k-1,i=0,1,\dots ,4k-1\}$$

$$\cup \{{[i,3k-1-r+i,3k+1+r+i]}_1\mid r=0,1,\dots ,k-2,i=0,1,\dots ,4k-1\}.$$

Notice that the subset of this set that results by restricting the value of i to 0 is a set of $T_1$s in which each edge difference and each arc difference modulo v appear exactly once, except for edge difference $2k=v/2$ and arc difference $3k$, which are absent. Therefore, we have

$$E\left(L\right)=\left\{\right(i,2k+i)\mid i=0,1,\dots ,2k-1\}$$

and

$$A\left(L\right)=\left\{\right(i,3k+i)\mid i=0,1,\dots ,4k-1\}.$$

Hence, $\left|E\right(L\left)\right|=2k=v/2$ and $\left|A\right(L\left)\right|=4k=v$, so that by Lemma 1, the packing is maximal.

Case 2. Suppose $v\equiv 2$ (mod 4), say with $v=4k+2$ where $k\ge 1$. Consider the collection of $T_1$s:

$$\{{[i,k-r+i,k+r+1+i]}_1,{[i,3k-r+i,3k+2+r+i]}_1\mid r=0,1,\dots ,k-1,i=0,1,\dots ,4k+1\}.$$

Notice that the subset of this set that results by restricting the value of i to 0 is a set of $T_1$s in which each edge difference and each arc difference modulo v appear exactly once, except for edge difference $2k+1=v/2$ and arc difference $3k+1$, which are absent. Therefore, we have

$$E\left(L\right)=\left\{\right\{i,2k+1+i\}\mid i=0,1,\dots ,2k\}$$

and

$$A\left(L\right)=\left\{\right(i,3k+1+i)\mid i=0,1,\dots ,4k+1\}.$$

Hence, $\left|E\right(L\left)\right|=v/2$ and $\left|A\right(L\left)\right|=v$, so that by Lemma 1, the packing is maximal.

In both cases, since $M_v$ is self-converse and the converse of $T_1$ is $T_2$, there is a maximal packing of $M_v$ with $T_2$s such that the leave $L^{\prime }$ satisfies $\left|E\right(L^{\prime }\left)\right|=v/2$ and $\left|A\right(L^{\prime }\left)\right|=v$ (the leave of the $T_2$-packing is the converse of the leave for the $T_1$-packing), as claimed. □

A limitation of our work is that we do not consider sufficient conditions for a maximal $T_3$-packing of $M_v$, but we conjecture that the necessary conditions given in Lemma 2 are sufficient, as follows.

Conjecture 1.

In a maximal packing of $M_v$ with $T_3$, where v is even, the leave L satisfies the following:

1. 

$\left|E\right(L\left)\right|=v/6$ and $\left|A\right(L\left)\right|=v/3$ when $v\equiv 0$ (mod $6)$;

2. 

$\left|E\right(L\left)\right|=(v+4)/6$ and $\left|A\right(L\left)\right|=(v+4)/3$ when $v\equiv 2$ (mod $6)$;

3. 

$\left|E\right(L\left)\right|=(v+2)/6$ and $A\left(L\right)|=(v+2)/3$ when $v\equiv 4$ (mod $6)$.

In summary, we classified maximal $T_1$ and $T_2$ packings of the complete mixed graph in terms of the size of the leave; this is realized in Theorem 2. We gave necessary conditions for the existence of a maximal $T_3$ packing of the complete mixed graph in Lemma 2 and conjectured that these necessary conditions are also sufficient.

4. Covering the Complete Mixed Graph with Mixed Triples

In this section, we present necessary conditions for the existence of a minimal covering of the complete mixed graph with each of the mixed triples, $T_1$, $T_2$, and $T_3$. We prove that the necessary conditions are in fact sufficient for $T_1$ and $T_2$. We do not address sufficiency for minimal $T_3$-coverings, but we conjecture that the necessary conditions we give for these are also sufficient.

Lemma 3.

In a covering of $M_v$ with $T_1$ where v is even, the padding consists of at least $v/2$ edges and at least v arcs. The same holds for coverings of $M_v$ with $T_2$.

Proof. 

For even v, each vertex of $M_v$ is of an out-degree $v-1$, which is odd. Since in $T_1$ each vertex is of an even out-degree, in any covering of $M_v$ with $T_1$, the padding is some mixed graph in which each vertex is of an odd out-degree. That is, each vertex of the padding is of an out-degree of at least one. So, the padding must contain at least v arcs. Since both $M_v$ and $T_1$ contain twice as many arcs as edges, this must be a property of the padding of a covering as well. Hence, the padding must contain at least $v/2$ edges and at least v arcs. The same argument holds for $T_2$, where “out-degree” is replaced with “in-degree”. □

In Theorem 3 below, we show that the lower bounds given in Lemma 3 are, in fact, attained in a minimal covering.

Lemma 4.

In a covering of $M_v$ with $T_3$ where v is even, the padding consists of the following:

1. 

At least $v/6$ edges and at least $v/3$ arcs when $v\equiv 0$ (mod $6)$;

2. 

At least $(v+4)/6$ edges and at least $(v+4)/3$ arcs when $v\equiv 2$ (mod $6)$;

3. 

At least $(v+2)/6$ edges and at least $(v+2)/3$ arcs when $v\equiv 4$ (mod $6)$.

Proof. 

For even v, each vertex of $M_v$ is of a total degree $3(v-1)$, which is odd. Since in $T_3$, each vertex is of an even total degree, in any covering of $M_v$ with $T_3$, the padding is some mixed graph in which each vertex is of an odd total degree. That is, each vertex of the padding is of a total degree of at least one. So, there must be at least $v/2$ edges and arcs total in the padding. Since both $M_v$ and $T_3$ contain twice as many arcs as edges, this must be a property of the padding of a covering as well. So, the padding must have a total number of edges and arcs that is a multiple of three. That is, the total number of edges is at least $\lceil v/6\rceil$, and the total number of arcs in the padding is at least $2\lceil v/6\rceil$. For $v\equiv 0$ (mod 6), we have $\lceil v/6\rceil =v/6$ and $2\lceil v/6\rceil =v/3$. For $v\equiv 2$ (mod 6), we have $\lceil v/6\rceil =(v+4)/6$ and $2\lceil v/6\rceil =(v+4)/3$. For $v\equiv 4$ (mod 6), we have $\lceil v/6\rceil =(v+2)/6$ and $2\lceil v/6\rceil =(v+2)/3$. So, the claimed lower bounds are established. □

Theorem 3.

In a minimal covering of $M_v$ with $T_1$, where $v\ge 3$, the padding P satisfies the following:

1. 

$E\left(P\right)=A\left(P\right)=⌀$ for v odd;

2. 

$\left|E\right(P\left)\right|=v/2$ and $\left|A\right(P\left)\right|=v$ for v even.

The same holds for minimal coverings of $M_v$ with $T_2$.

Proof. 

If v is odd, then a $T_1$-decomposition and a $T_2$-decomposition of $M_v$ exist by Theorem 1 so that $E\left(P\right)=A\left(P\right)=⌀$ in this case, as claimed.

Case 1. Suppose $v\equiv 0$ (mod 4), say with $v=4k$ where $k\ge 1$. Consider the collection of $T_1$s:

$$\{{[i,k-r+i,k+r+1+i]}_1\mid r=0,1,\dots ,k-1,i=0,1,\dots ,4k-1\}$$

$$\cup \{{[i,3k-1-r+i,3k+1+r+i]}_1\mid r=0,1,\dots ,k-2,i=0,1,\dots ,4k-1\}$$

$$\cup \{{[i,k+i,3k+i]}_1\mid i=0,1,\dots ,4k-1\}.$$

Notice that the subset of this set that results by restricting the value of i to 0 is a set of $T_1$s in which each edge difference and each arc difference modulo v appear exactly once, except for edge difference $2k=v/2$ and arc difference k, each of which appears twice. Therefore, we have

$$E\left(P\right)=\left\{\right\{i,2k+i\}\mid i=0,1,\dots ,2k-1\}$$

and

$$A\left(P\right)=\left\{\right(i,k+i)\mid i=0,1,\dots ,4k-1\}.$$

Hence, $\left|E\right(P\left)\right|=v/2$ and $\left|A\right(P\left)\right|=v$, so that by Lemma 3, the covering is minimal.

Case 2. Suppose $v\equiv 2$ (mod 4), say with $v=4k+2$ where $k\ge 1$. Consider the collection of $T_1$s:

$$\{{[i,k-r+i,k+r+1+i]}_1,{[i,3k-r+i,3k+2+r+i]}_1\mid r=0,1,\dots ,k-1,i=0,1,\dots ,4k+1\}$$

$$\cup \{{[i,k+i,3k+1+i]}_1\mid i=0,1,\dots ,4k-1\}.$$

Notice that the subset of this set that results by restricting the value of i to 0 is a set of $T_1$s in which each edge difference and each arc difference modulo v appear exactly once, except for edge difference $2k+1=v/2$ and arc difference k, each of which appears twice. Therefore, we have

$$E\left(P\right)=\left\{\right\{i,2k+1+i\}\mid i=0,1,\dots ,2k\}$$

and

$$A\left(P\right)=\left\{\right(i,k+i)\mid i=0,1,\dots ,4k-1\}.$$

Hence, $\left|E\right(P\left)\right|=v/2$ and $\left|A\right(P\left)\right|=v$, so that by Lemma 3, the covering is minimal.

In both cases, since $M_v$ is self-converse and the converse of $T_1$ is $T_2$, there is a minimal covering of $M_v$ with $T_2$s such that the padding $P^{\prime }$ satisfies $\left|E\right(P^{\prime }\left)\right|=v/2$ and $\left|A\right(P^{\prime }\left)\right|=v$ (the padding of the $T_2$ covering is the converse of the padding for the $T_1$ covering), as claimed. □

A limitation of our work is that we do not consider sufficient conditions for a minimal $T_3$ covering of $M_v$, but we conjecture that the necessary conditions given in Lemma 4 are sufficient, as follows.

Conjecture 2.

In a minimal covering of $M_v$ with $T_3$ where v is even, the padding P satisfies the following:

1. 

$\left|E\right(P\left)\right|=v/6$ and $\left|A\right(P\left)\right|=v/3$ when $v\equiv 0$ (mod $6)$;

2. 

$\left|E\right(P\left)\right|=(v+4)/6$ and $\left|A\right(P\left)\right|=(v+4)/3$ when $v\equiv 2$ (mod $6)$;

3. 

$\left|E\right(P\left)\right|=(v+2)/6$ and $A\left(P\right)|=(v+2)/3$ when $v\equiv 4$ (mod $6)$.

In summary, we classified minimal $T_1$ and $T_2$ coverings of the complete mixed graph in terms of the size of the padding; this is realized in Theorem 3. We have given necessary conditions for the existence of a minimal $T_3$ covering of the complete mixed graph in Lemma 4 and conjectured that these necessary conditions are also sufficient.

5. Discussion

In this paper, we extended the idea of packings and coverings to mixed graphs. We classified minimal coverings and maximal packings of the complete mixed graph with the mixed triples $T_1$ and $T_2$. We gace necessary conditions for $T_3$ minimal coverings and maximal packings of the complete mixed graph and conjectured that these conditions are also sufficient.

Graph decompositions in general have numerous applications, including coding theory, radio astronomy, communication networks, and experimental design [14]. Although the idea of graph decomposition goes back well over 150 years in the form of Steiner triple systems, the study of mixed graph decompositions goes back only 25 years. Mixed graphs can be used to model, for example, a network of roads, some of which are one-way roads (represented by arcs) and others that are two-way roads (represented by edges). Network theory currently incorporates graphs and digraphs (see [15]) but has yet to capitalize on the potential of mixed graphs. We gave some possible agricultural applications of mixed graphs related to crop rotation and companion planting. Another possible agricultural application involves terraced gardening or planting crops in terraced fields. In this setting, we have nutrients washing downhill so that beneficial nutrients produced by one plant (nitrogen, for example) may wash into the next field downhill. This situation allows for one crop to influence another but not conversely (represented by an arc), whereas crops planted on the same level (represented by an edge) can produce the mutual benefits of companion planting described in Section 1.6. This allows for a one-way interaction imposed by the spatial situation, as opposed to the temporal imposition of this interaction, as described above. Mixed graphs can be used to describe these interactions in any setting where there is the possibility for one-way and two-way interactions, whether the influence involves spatial or temporal (or other) separation. As another example, one could compare student performance in various classes when they take certain classes at the same time or not. We might expect a student who has had calculus before taking physics to perform better in the physics class, or we might expect a student who has already taken physics to perform better in calculus. On the other hand, a student might take the two classes together and benefit from the fact that related ideas are covered in both classes. We could consider students who take world history before European history (or vice versa), and/or a foreign language at the same time, etc. As a final and more general example of possible applications of mixed graphs, we consider the situation of a model involving an explanatory variable and a response variable. In particular, if the explanatory variable involves ordered categories (such as an economic system with lower, middle, and upper classes), then a comparison of two populations involves three possibilities: (1) the first population is in a higher category than the second, (2) the first population is in a lower category than the second, and (3) the two populations are in the same category. These three possibilities can be represented by treating the populations as vertices and the comparisons of populations by arcs from lower to higher category populations or by edges when two populations are in the same category. As the categorical variables vary over the ordered categories, the complete mixed graph reflects all possible configurations of interest. In such a general setting, the response variable is measured or computed over all arcs and edges of the complete mixed graph. Mixed graphs then have potential applications in the design of such experiments.

In terms of mixed graphs decompositions as a mathematical structure of study, there is still much to accomplish. The conjectured $T_3$ minimal covering and maximal packing problems are the most relevant to the present work. Hybrid mixed triple systems, where $M_v$ is to be decomposed into some $T_1$s, some $T_2$s, and some $T_3$s (as mentioned in Section 1.6) are another topic; these are decompositions of $M_v$, but not isomorphic decompositions. In addition, designs (that is, decompositions of $M_v$ into other small mixed graphs) present a number of open problems. Decompositions, packings, and coverings of the $\lambda$-fold complete mixed graph (where each edge and each arc is repeated $\lambda$ times) are open problems. The complete mixed graph with a hole, in which the edges and arcs of a complete mixed subgraph of $M_v$ are removed from $M_v$, can be used to model certain desirable design theory problems. Decompositions, packings, coverings of the complete mixed graph with a hole provide numerous open problems (in addition to the corresponding $\lambda$-fold versions of these problems).