Various Theorems on Tournaments
Abstract
In this thesis we prove a variety of theorems on tournaments. A prime tournament is a tournament such that there is no , , such that for every vertex , either for all or for all . First, we prove that given a prime tournament which is not in one of three special families of tournaments, for any prime subtournament of with there exists a prime subtournament of with vertices that has a subtournament isomorphic to . We next prove that for any two cyclic triangles , in a prime tournament , there is a sequence of cyclic triangles such that , , and shares an edge with for all . Next, we consider what we call matching tournaments, tournaments whose vertices can be ordered in a horizontal line so that every vertex is the head or tail of at most one edge that points righttoleft. We determine the conditions under which a tournament can have two different orderings satisfying the above conditions. We also prove that there are infinitely many minimal tournaments that are not matching tournaments. Finally, we consider the tournaments and , which are obtained from the transitive tournament with vertices by reversing the edge from the second vertex to the last vertex and from the first vertex to the secondtolast vertex, respectively. We prove a structure theorem describing tournaments which exclude and as subtournaments.
Acknowledgments
None of this work would have been possible without my adviser Paul Seymour, and this thesis very much reflects his ideas, guidance, and contributions. I would also like to thank Maria Chudnovsky for putting in the time and work to be my second reader.
1 Introduction
A tournament is a nonnull, loopless directed graph such that for any two distinct vertices , there is exactly one edge with both ends in . A subtournament of a tournament is a tournament induced on a nonempty subset of . If is a subtournament of and , we use to denote the subtournament of induced on . If , we use to denote the subtournament induced on . We use to mean and to mean . For each vertex , let be the set of outneighbors of in , and let be the set of inneighbors of in . We call the outdegree of in and the indegree of in . For any two disjoint sets , we write if for every and . We use to mean . Given distinct vertices , define so that if and if . Finally, an ordering of is a list of the vertices of . A transitive tournament is a tournament whose vertices can be ordered such that if . We call the unique ordering which satisfies the previous condition the standard ordering of the vertices of a transitive tournament. We use to denote the isomorphism class of transitive tournaments with vertices. We will often refer to as a tournament itself; likewise, when we define other isomorphism classes of tournaments, we will often refer to them as tournaments themselves.
This thesis contains various results on tournaments proven over the course of the year. While the different major results are largely independent of each other, there are a few common ideas. Section 2 introduces homogeneous sets, the “substitution” construction, and prime tournaments, concepts that are central to this paper and will be used throughout. Sections 3 through 6 each showcase a different theorem. Section 3 gives a theorem that allows us to “grow” a prime tournament one vertex at a time starting from any of its prime subtournaments. This result strengthens a theorem by Schmerl and Trotter [8] which is often used in the study of prime tournaments (see for example [1], [2], [6]). In Section 4 we prove an interesting result on the structure of cyclic triangles within prime tournaments. Sections 5 and 6 deal with tournaments which are in a sense “almosttransitive”—they are formed from transitive tournaments by reversing a set of edges satisfying a given property. Section 5 deals with matching tournaments, tournaments for which this set of edges is a matching. Section 6 deals with tournaments with only a single edge reversed, and considers the structure of tournaments that exclude these. The final section considers directions for future research.
While this thesis is meant to be read chronologically, Sections 3 through 6 are more or less independent of each other with the following exceptions: the proofs of Theorems 4.1 and 6.2 use the concept of weaves defined in the proof of Theorem 3.1, and Section 6 uses the definition of a backedge given at the beginning of Section 5.
2 Homogeneous sets and prime tournaments
2.1 Basic definitions and properties
Given a tournament , a homogeneous set of is a subset of vertices such that for all vertices , either or . A homogeneous set is nontrivial if ; otherwise it is trivial.
We list some basic properties of homogeneous sets.
Proposition 2.1 (Restriction).
If is a homogeneous set of a tournament , and is a subtournament of , then is a homogeneous set of .
Proposition 2.2 (Extension).
If , are subtournaments of a tournament and is a homogeneous set of both and , then is a homogeneous set of the tournament induced on .
Proposition 2.3 (Cloning).
Let be a tournament, be distinct vertices, and . If is a homogeneous set of and is a homogeneous set of , then is a homogeneous set of .
Proposition 2.4 (Intersection).
If , are homogeneous sets of a tournament , then is a homogeneous set of .
Proposition 2.5 (Subtraction).
If , are homogeneous sets of a tournament and , then is a homogeneous set of .
Proposition 2.6 (Union).
Suppose is a tournament and such that , is a homogeneous set of , and is a homogeneous set of . Then is a homogeneous set of .
A tournament is prime if all of its homogeneous sets are trivial; otherwise, it is decomposable. Given a tournament and an ordering of its vertices, and given tournaments , let be a tournament with vertex set , where the are pairwise disjoint sets of vertices, such that if , and for all the subtournament of induced on is isomorphic to . Every tournament with at least two vertices can be written as where is a prime tournament with at least two vertices. The prime tournaments are precisely those tournaments which cannot be written as for some with .
A tournament is strongly connected if for any two vertices , there is a directed path from to and a directed path from to . A strongly connected component, or strong component, of a tournament is a maximal strongly connected subtournament of . The strong components of a tournament can be ordered as so that can be written as , where has the standard ordering for a transitive tournament. From this we see that the vertices of a strong component of a tournament form a homogeneous set, so if a tournament is prime, either it is strongly connected or all of its strong components have only one vertex. In the latter case the tournament is transitive, and has a homogeneous set for . So every prime tournament with vertices is strongly connected.
The tournament with 1 vertex and the tournament with 2 vertices are both prime. The only prime tournament with 3 vertices is the cyclic triangle. It can be checked that all tournaments with 4 vertices are decomposable. For 5 vertices, there are exactly three prime tournaments , , and , drawn below.
These three tournaments can be generalized to any odd number of vertices as follows.
Definition 2.7.
Let and . Define the tournaments , , and as follows.

is the tournament with vertices such that if or .

is the tournament obtained from by reversing all edges which have both ends in .

is the tournmanet with vertices such that if , and .
Note the following degenerate cases: , , and are all the singlevertex tournament, and , , and are all the cyclic triangle.
It can checked that , , and are prime for all odd . Also, note that , , and have subtournaments isomorphic to , , and , respectively, for . In fact, the only prime subtournaments of , , and with at least 3 vertices are , , and , respectively, for .
2.2 Prime subtournaments of prime tournaments
Given a prime tournament, it is natural to ask about its prime subtournaments. There have been several results proven about prime subtournaments; to state them, we introduce some extra terminology. Let be a tournament and be a prime subtournament of with . Define to be the set of vertices for which the tournament induced on is prime. Let be the set of vertices for which is a homogeneous set of . Finally, for each , let be the set of vertices for which is a homogeneous set of .
We can use the basic properties of homogeneous tournaments found in Propositions 2.1 through 2.6 to prove the following, which appears in [4].
Proposition 2.8 (Ehrenfeucht and Rozenberg [4]).
Let be a tournament and be a prime subtournament of with . Then

The collection of sets forms a partition of .

If , , such that is decomposable, then is a homogeneous set of .

If and such that is decomposable, then is a homogeneous set of .

If for some and such that is decomposable, then is a homogeneous set of .
Proof.
We will only prove part (i); the proofs of the other parts involve similar techniques. We need to show that if is a vertex such that is decomposable, then is in exactly one of the sets , for . First, we show that is in at least one of these sets. Since is decomposable, it has a nontrivial homogeneous set . By restriction to , we have that is a homogeneous set of . Since is prime, we must thus have either or . If the former case, then since is a nontrivial homogeneous set of , we must have and , so , as desired. If the latter case, then , as desired. So is in at least one of , for .
Suppose is in at least two of these sets. First, suppose and where . We have that and are homogeneous sets of . Applying Proposition 2.5, we have that is a homogeneous set of , and restricting to we have that is a homogeneous set of . Since , this is a nontrivial homogeneous set of , a contradiction.
Now suppose and for two distinct . Then and are homogeneous sets of . Applying Proposition 2.6, we have that is a homogeneous set of , and restricting to , we have that is a homogeneous set of . Since , this is a nontrivial homogeneous set of , a contradiction. This completes the proof. ∎
This result leads to the following corollaries in the case where is prime.
Corollary 2.9 (Ehrenfeucht and Rozenberg [4]).
Let be a prime tournament and be a prime subtournament of with . Then there exist distinct vertices such that is prime.
Proof.
Suppose that for every distinct , is decomposable. We first prove that and are empty for all . By Proposition 2.8(iii), we have that is a homogeneous set of for every and 2.2 and 2.6 on these homogeneous sets, we thus have that is a homogeneous set of . If is nonempty, then is a nontrivial homogeneous set of , contradicting the fact that is prime. So is empty. . Repeatedly applying Propositions
Similarly, for each , we have by Proposition 2.8(iv) that is a homogeneous set of for every , 2.2 and 2.6 on these homogeneous sets, we have that is a homogeneous set of . If is nonempty, then is a nontrivial homogeneous set of , a contradiction. So is empty for each . . Repeatedly applying Propositions
Corollary 2.10.
Every prime tournament with has a prime subtournament with 5 vertices.
Proof.
Since is prime and has vertices, it is not transitive, so it contains a cyclic triangle. Applying Corollary 2.9 with as a cyclic triangle gives the corollary. ∎
Corollary 2.9 can be thought of as a “growing” lemma: within a prime tournament, we can grow an increasing sequence of prime subtournaments so that each subtournament contains the previous one. For example, starting with a cyclic triangle in a prime tournament and repeatedly applying the corollary, we have that contains a prime subtournament with vertices for every odd . In particular, contains a prime subtournament with either or vertices. This last statement was improved upon by Schmerl and Trotter in [8].
Theorem 2.11 (Schmerl and Trotter [8]).
If is a prime tournament with , and is not , , or for any odd , then has a prime subtournament with vertices.
Theorem 2.12 (Schmerl and Trotter [8]).
If is a prime tournament with , then has a prime subtournament with vertices.
Schmerl and Trotter’s proof of these two theorems involve “growing” prime subtournaments, during which Corollary 2.9 is essential. Our result in the next section can be thought of as a strengthening of Corollary 2.9 in that it allows us to grow the sequence of prime subtournaments one vertex at a time instead of two vertices at a time in the case where is not , , or . In particular, our theorem has both Theorems 2.11 and 2.12 as immediate corollaries.
3 Growing prime tournaments
3.1 Statement of theorem
The main theorem is as follows.
Theorem 3.1.
Let be a prime tournament which is not , , or for any odd , and let be a prime subtournament of with . Then there exists a prime subtournament of with vertices that has a subtournament isomorphic to .
Note that Theorem 3.1 is not a strict strengthening of Corollary 2.9 because the theorem does not guarantee that the subtournament with vertices contains the actual vertices of ; it only guarantees that it contains a subtournament isomorphic to . However, in many applications where one would want to grow prime subtournaments, only the isomorphism class of the previous subtournament matters; for example, to use this theorem to prove Theorems 2.11 and 2.12, only the number of vertices at each step matters.
Theorem 3.1 is based on and is a direct analogue of a theorem by Chudnovsky and Seymour for undirected graphs, found in [3]. Indeed, the proof for Thoerem 3.1 found here is closely related to the proof found in [3]. Chudnovsky and Seymour used their theorem to develop a polynomialtime algorithm to find simplicial cliques in prime clawfree graphs.
3.2 Proof of theorem
Or proof consists of two main claims.
Claim 1.
The theorem holds when .
Claim 2.
If is odd and is a prime tournament with vertices that has a subtournament isomorphic to (, , respectively), then has a prime subtournament with vertices that has a subtournament isomorphic to (, , respectively).
Assuming the truth of these two claims, we can prove Theorem 3.1 as follows: By Claim 1, we can assume . First suppose that is not , , or for any . By Corollary 2.9, there are vertices such that is prime. Since is not , , or for any , is not any of these tournaments either (see the second paragraph after Definition 2.7). So applying Claim 1 to with subtournament , we have a subtournament of with vertices that has a subtournament isomorphic to , as desired.
Now assume is , , or for some odd . We will assume is ; the arguments for and are identical. Suppose there is no prime subtournament of with vertices that has a subtournament isomorphic to . Let be the largest odd integer such that

has a subtournament isomorphic to , and

has no prime subtournament with vertices that has a subtournament isomorphic to .
Thus, . By Claim 1, we have . We claim that has a subtournament isomorphic to . Applying Corollary 2.9 on a copy of in , we have that has a prime subtournament with vertices that has a subtournament isomorphic to . If is not , then we can apply Claim 1 to it to obtain a prime subtournament with vertices that has a subtournament isomorphic to , contradicting the definition of . Thus, is .
Thus, has a subtournament isomorphic to . Hence, by the maximality of , there is a prime subtournament of with vertices that has a subtournament isomorphic to . Applying Claim 2, has a prime subtournament with vertices that has a subtournament isomorphic to . This contradicts the definition of , completing the proof.
We now prove the two claims.
Proof of Claim 1.
The proof of this claim is made considerably simpler by establishing the right definitions, so we will spend a good amount of time doing so. These definitions will also be useful in later proofs.
We define a weave to be a tournament with vertices such that

if and have opposite parity

One of the following holds:

for all with odd

for all with odd


One of the following holds:

for all with even

for all with even

Using “F” to mean “forward” and “B” to mean “backward,” we will call a weave an FF weave if (2a) and (3a) hold, an FB weave if (2a) and (3b) hold, a BF weave if (2b) and (3a) hold, and a BB weave if (2b) and (3b) hold. We refer to these as the four types of weaves. (A weave can be of more than one type if .)
The following facts can be easily checked; the information that is most relevant to our proof is summarized in the corollary afterwards.
Proposition 3.2.
Let be a weave.

If is FF, then it is transitive.

If is even and is FB, then is a homogeneous set of .

If is even and is BF, then is a homogeneous set of .

If is even and is BB, then is a homogeneous set of .

If is odd and is FB, then is a homogeneous set of .

If is odd and is BF, then is .

If is odd and is BB, then is .
Let be a vertex such that for odd and for even .

If is odd and is FF or FB, then is a homogeneous set of .

If is odd and is BF or BB, then is a homogeneous set of .

If is even and is FF, then is .

If is even and is FB or BF, then is .

If is even and is BB, then is .
Corollary 3.3.
Let be a weave.

If and is prime, then is or for some .

If and is a vertex as in Proposition 3.2, and is prime, then is , , or for some .
The following fact will also be important.
Proposition 3.4.
If is a weave and , then is a weave of the same type as .
Corollary 3.5.
If is a weave, then the subtournaments between and such that if , then and have the same parity. are all isomorphic to each other. Furthermore, there is an isomorphism
We are now ready to prove the claim. The case is trivial, so assume . Suppose there is no prime subtournament of with vertices that has a subtournament isomorphic to . Let , where . We will call a weave a weave if all of the following hold:

is a subtournament of .

and for some .

is a homogeneous set in .

is a homogeneous set in .
Since is a weave, at least one weave with vertices exists. Let be a weave which maximizes . If , then by Corollary 3.3 is or , contradicting the assumptions of the theorem. So .
Now, by Corollary 3.5, is isomorphic to for all . In fact, because of the second sentence of Corollary 3.5 and (c) and (d) in the definition of a weave, we have that is isomorphic to for all .
Let . Since is isomorphic to , by our original assumption must be decomposable. It follows that either or for some . Suppose for some . We thus have

, and

if , then if is FF or BF and if is FB or BB.
First suppose that . Since is a homogeneous set of but it is not a homogeneous set of , and , we must have . Thus, . Furthermore, by (d) in the definition of a weave, we have that if and if . Finally, by Proposition 2.6, is a homogeneous set in . Thus, is a weave. This contradicts the maximality of , so .
Now suppose . We cannot have for any even because but for all even . So for some odd . Now, since , for every we have . Thus, since is odd, from (c) and (d) in the definition of a weave we have that is a homogeneous set of . Since as mentioned earlier, this contradicts the primeness of .
Thus, we cannot have for any . So . By symmetry, we also have that , where . If , then we have , and hence is a homogeneous set of , a contradiction since . So assume . Then since and , we have . Similarly since and , we have . In particular, we have . If is odd, then (c) and (d) from the definition of a weave imply that , a contradiction since is nonempty. So is even, and (c) and (d) imply that
Thus, is a homogeneous set of , and as we showed before it is nonempty. Hence, . Let . Then , and by Corollary 3.3, must be , , or . This is a contradiction, completing the proof of Claim 1. ∎
Proof of Claim 2.
We first prove the following general proposition.
Proposition 3.6.
Let be a prime tournament and let . Let , be prime subtournaments of such that , and are decomposable, and the subtournament induced on is prime with . Then either for some or for some .
Proof.
Since is prime and is decomposable, we have either or for some , and likewise for .
First, suppose and . Then is a homogeneous set of and is a homogeneous set of . Since by assumption, by Proposition 2.6 we have that is a homogeneous set of . This contradicts the primeness of , so we cannot have this case.
Next, suppose and for some . If then we are done, so assume , and hence . Now, is a homogeneous set of and is a homogeneous set of ; restricting to the subtournament , we have that and are homogeneous sets of . Applying Proposition 2.5, we have that is a homogeneous set of , and hence is a homogeneous set of . Since , this contradicts the primeness of .
Finally, suppose for some and for some . If either or then we are done, so assume . If , then is a homogeneous set of both and , so by Proposition 2.2, is a homogeneous set of , a contradiction. If , then restricting to and applying Proposition 2.6, we have that is a homogeneous set of , so is a homogeneous set of . As before, this is a contradiction, which completes the proof. ∎
We can now prove Claim 2. Assume the hypotheses of Claim 2. Let be a subtournament of which is isomorphic to , , or . Let and be prime subtournaments of with and vertices, respectively. In other words, if is isomorphic to , then is isomorphic to and is isomorphic to ; likewise for and . We wish to prove there is a prime subtournment of with vertices that has a subtournament isomorphic to .
Suppose the contrary. Let . By Proposition 3.2, we can write as , where is a weave and is as in Proposition 3.2. For distinct integers , let . Note that is isomorphic to for all . Hence, by assumption, must be decomposable for all .
Now, for distinct integers , we define as follows: If for some , then let ; otherwise, let . By Proposition 2.8(i), is welldefined. Now, suppose are integers such that . Then is isomorphic to , and is hence prime. Thus, applying Proposition 3.6 with as and as , we have that for all such , either or . We will denote this fact as ().
Applying () with and , we have that either or . Without loss of generality, assume
Now, applying () with and , we have that either or . Since and , we must have
Finally, applying () with and , we have that either or . Since we already have , we must have
Now, by the definition of , we have that , , and are homogeneous sets of , , and , respectively. Restricting to the subtournament (the equality holds because ), we have that and are homogeneous sets of , and hence by Proposition 2.6 and restriction, we have that is a homogeneous set of . Similarly,