Tournament (graph theory)

Tournament
A tournament on 4 vertices
Vertices	$n$
Edges	${\binom {n}{2}}$
Table of graphs and parameters

A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph.

The name tournament originates from such a graph's interpretation as the outcome of a round-robin tournament in which every player encounters every other player exactly once, and in which no draws occur. In the tournament digraph, the vertices correspond to the players. The edge between each pair of players is oriented from the winner to the loser. If player $a$ beats player $b$, then it is said that $a$ dominates $b$.

Paths and cycles

Any tournament on a finite number $n$ of vertices contains a Hamiltonian path, i.e., directed path on all $n$ vertices (Rédei 1934). This is easily shown by induction on $n$: suppose that the statement holds for $n$, and consider any tournament $T$ on $n+1$ vertices. Choose a vertex $v_{0}$ of $T$ and consider a directed path $v_{1},v_{2},\ldots ,v_{n}$ in $T\setminus \{v_{0}\}$. Now let $i\in \{0,\ldots ,n\}$ be maximal such that $v_{j}\rightarrow v_{0}$ for all $j$ with $1\leq j\leq i$. Then

$v_{1},\ldots ,v_{i},v_{0},v_{i+1},\ldots ,v_{n}$

is a directed path as desired.

Further, any strongly connected tournament on $n\geq 3$ vertices contains cycles of length $3,\dots ,n$. As a corollary, a tournament on n vertices is strongly connected if and only if every vertex is on a cycle of length k for $3\geq k\geq n$. This implies that a strongly connected tournament has a Hamiltonian cycle (Camion 1959). Moreover, if the tournament is 4‑connected, each pair of vertices can be connected with a Hamiltonian path (Thomassen 1980).

Transitivity

A transitive tournament on 8 vertices.

A tournament in which $(a\rightarrow c$ and $b\rightarrow c)$ $\Rightarrow$ $(a\rightarrow c)$ is called transitive.

A player who wins all games would naturally be the tournament's winner. However, as the above example shows, there might not be such a player. A tournament for which there isn't is called a $1$-paradoxical tournament. More generally, a tournament $T=(V,E)$ is called $k$-paradoxical if for every $k$-subset $V'$ of $V$ there is a $v_{0}\in V\setminus V'$ such that $v_{0}\rightarrow v$ for all $v\in V'$. By means of the probabilistic method Paul Erdős showed that if $\vert V\vert$ is sufficiently large, then almost every tournament on $V$ is $k$-paradoxical.

Number of possible results

Let us have $n$ competitors, and a scoring system where no ties are permitted. Let $s_{i}$ be the number of wins scored by competitor $i$, and let us order them such that $s_{1}\leq s_{2}\leq \cdots \leq s_{n}$. Thus we define a score vector, $(s_{1},s_{2},\cdots ,s_{n})$, satisfying the following three conditions:

1. $0\leq s_{1}\leq s_{2}\leq \cdots \leq s_{n}$
2. $s_{1}+s_{2}+\cdots +s_{i}\geq {i \choose 2},{\mbox{for }}i=1,2,\cdots ,n-1$
3. $s_{1}+s_{2}+\cdots +s_{n}={n \choose 2}$

Let $s(n)$ be the number of different score vectors of size $n$. The sequence $s(n)$ (sequence A000571 in the OEIS) starts as:

1, 1, 1, 2, 4, 9, 22, 59, 167, 490, 1486, 4639, 14805, 48107, ...

It is possible to show that for sufficiently large n:

$s(n)>c_{1}4^{n}n^{-{5 \over 2}}$

Also:

$s(n)<c_{2}4^{n}n^{-{5 \over 2}}$

Where: $c_{1}=0.049,\,\,c_{2}<4.858.$

Together these show that:

$s(n)\in \Theta (4^{n}n^{-{5 \over 2}}).$

Here $\Theta$ signifies an asymptotically tight bound.

References

• Rédei, László (1934), "Ein kombinatorischer Satz", Acta Litteraria Szeged, 7: 39–43{{citation}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)
• Camion, Paul (1959), "Chemie et circuits hamiltoniens des graphes complets", Comptes Rendus de l'Académie des Sciences de Paris, 249: 2151–2152
• Thomassen, Carsten (1980), "Hamiltonian-Connected Tournaments", Journal of Combinatorial Theory, Series B, 28: 142–163, doi:10.1016/0095-8956(80)90061-1
• Takacs, Lajos (1991). "A Bernoulli Excursion and Its Various Applications". Advances in Applied Probability. 23 (3): 557–585. doi:10.2307/1427622.

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tournament at PlanetMath.