A Characterization of Unique Tournaments

Journal of Combinatorial Theory, Series B  TB1799 Journal of Combinatorial Theory, Series B 72, 157159 (1998) Article No. TB971799

NOTE A Characterization of Unique Tournaments Prasad Tetali School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160 E-mail: tetalimath.gatech.edu Received September 20, 1995 We call a tournament unique, if there is no other tournament (barring isomorphic ones) which shares the same score vector. In this note, we provide a simple characterization of such unique tournaments.  1998 Academic Press

A tournament is a complete oriented graph. The score vector of a tournament is simply the sequence of outdegrees of the vertices of the tournament. Let T n denote a tournament on n vertices, and S(T n ) denote the score vector of T n . Thus S(T n )=(s 1 , ..., s n ), where 0s i n&1,  i s i =( n2 ), and we assume s 1 s 2  } } } s n . In general, there can be many non-isomorphic tournaments which share the same score vector. However, we shall focus on the special class (which we call Unique) consisting of those score vectors for which there is a unique tournament up to isomorphism. The score vectors (1, 1, 1), (0, 1, 2, 3), and (2, 2, 2, 2, 2) are some obvious examples. In general, any transitive tournament on n vertices (which has the score vector (0, 1, ..., n&1)) is also an example. More formally speaking, let T n tT$n denote the fact that T n is isomorphic to T $n . Let Un =[T n : S(T n )=S(T$n ) O T n tT $n ]. We define Unique=  n=1 Un . In this note we report a complete characterization of the class Unique. The characterization is such that it gives us the following enumeration result. Let u n denote the cardinality of Un . That is, u n is the number of score vectors (or tournaments) on n vertices with the above-mentioned property. We show that u 1 =1, u 2 =1, u 3 =2, u 4 =4, and u 5 =7, and for all n>5, u n =u n&1 +u n&3 +u n&4 +u n&5 . This result is similar in spirit to a result of Garey [2], wherein the number of tournaments on n vertices (n4) having exactly one Hamiltonian cycle was shown to be F 2n&6 , the (2n&6)th Fibonacci number (see also [1]). A tournament is strong if it is strongly connected in the sense of a digraph. Theorem 1. There are exactly four (basic) strong tournaments in Unique (see Fig. 1); any other (nonstrong) tournament in Unique can be decomposed into strong components, each of which is one of the four basic tournaments. 157 0095-895698 25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.

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NOTE

Fig. 1.

The four basic unique tournaments.

We make use of two existing results (Theorem 2 and Theorem 3 below) to prove the above theorem. We begin with the definition of a simple tournament (see [4] for information on simple tournaments). Let T n = (V(T n ), A(T n )) represent a tournament on n vertices. If (w, z) # A(T n ) then we say w beats z (and z is beaten by w). Definition 1. T n is simple if, for any proper subset M of V(T n ), there exists a z # V(T n )"M such that z beats at least one vertex in M and z is beaten by at least one vertex in M. A simple tournament is clearly strong, but the converse is not necessarily truethe strong tournament on four vertices, for example, is not simple. Definition 2. We call a score vector S forcibly simple (FS) if every tournament with the score vector S is simple. Muller et al. [4] prove, inter alia, the following interesting theorems. We paraphrase the theorems to suit our terminology. Theorem 2 (Muller et al.). For each score vector S n which corresponds to a strong tournament on n{4 vertices, there is a simple tournament which has the same score vector. Theorem 3. (Muller et al.). The following two statements are equivalent: (1)

S is forcibly simple;

(2)

S # [(0), (0, 1), (1, 1, 1), (2, 2, 2, 2, 2), (3, 3, 3, 3, 3, 3, 3)].

Proof of Theorem 1. The proof is by drawing necessary conclusions from the above two theorems. Theorem 2 implies that a strong tournament T n , for n{4, which is unique (i.e., belongs to Un ) is (necessarily) simple, and its score vector is (trivially) forcibly simple. Now using Theorem 3, we

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can conclude that a score vector which corresponds to a tournament that is both strong and unique (except for n=4) must come from the list of five score vectors in statement (2) of Theorem 3. The score vector (0, 1) corresponds to a unique tournament which is not strong; it is easy to check that there are three nonisomorphic strong tournaments which have the score vector (3, 3, 3, 3, 3, 3, 3). (Note that this among other things gives us that there are no strong tournaments on six or more vertices which belong to Unique.) The only case not covered by the above is the unique strong tournament on four vertices with the score vector (1, 1, 2, 2). Thus the only strong tournaments which are in Unique are the four tournaments shown in Fig. 1. Furthermore, it is easy to see that the strong components of any nonstrong tournament in Unique are themselves members of Unique. Thus the cyclic (or strong) decomposition of any tournament in Unique consists of only the four basic tournaments shown in Fig. 1. K Remark. It is a basic exercise in recurrence relations to show that the above characterization yields for n6, u n =u n&1 +u n&3 +u n&4 +u n&5 , from which it follows that there exist constants cr0.48 and :r1.685 such that lim u n c: n =1.

n

We also have u 1 =1, u 2 =1, u 3 =2, u 4 =4, u 5 =7.

ACKNOWLEDGMENTS We thank Jeong Han Kim and Richard Duke for very valuable discussions and corrections. We are grateful to the anonymous referee who pointed out a mistake in an earlier version of this note.

REFERENCES 1. R. J. Douglas, Tournaments that admit exactly one Hamiltonian circuit, Proc. London Math. Soc. 3 (1970), 716730. 2. M. R. Garey, On enumerating tournaments that admit exactly one Hamiltonian circuit, J. Combin. Theory Ser. B 13 (1972), 266269. 3. J. W. Moon, ``Topics on Tournaments,'' Rinehart and Winston, New York, 1968. 4. V. Muller, J. Nes$ etr$ il, and J. Pelant, Either tournaments or algebras?, Discrete Math. 11 (1975), 3766. 5. K. B. Reid and L. W. Beineke, Tournaments, in ``Selected Topics in Graph Theory'' (L. W. Beineke and R. J. Wilson, Eds. ), Chap. 7, Academic Press, New York, 1978.

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