Cycles of Each Length in Regular Tournaments

1967 ◽  
Vol 10 (2) ◽  
pp. 283-286 ◽  
Author(s):  
Brian Alspach
Keyword(s):  
Strong Tournament ◽  
Regular Tournament

It is known that a strong tournament of order n contains a cycle of each length k, k=3,…, n, ([l], Thm. 7). Moon [2] observed that each vertex in a strong tournament of order n is contained in a cycle of each length k, k = 3,…, n. In this paper we obtain a similar result for each arc of a regular tournament, that is, a tournament in which all vertices have the same score.

scholarly journals An s-strong tournament with s⩾3 has s+1 vertices whose out-arcs are 4-pancyclic

2006 ◽  
Vol 154 (18) ◽  
pp. 2609-2612 ◽  
Author(s):  
Jinfeng Feng ◽  
Shengjia Li ◽  
Ruijuan Li
Keyword(s):  
Strong Tournament

scholarly journals Symmetry Breaking in Tournaments

10.37236/3182 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Antoni Lozano
Keyword(s):  
Symmetry Breaking ◽  
Upper Bounds ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Nontrivial Automorphism ◽  
Determining Set ◽  
Strong Tournament

We provide upper bounds for the determining number and the metric dimension of tournaments. A set of vertices $S \subseteq V(T)$ is a determining set for a tournament $T$ if every nontrivial automorphism of $T$ moves at least one vertex of $S$, while $S$ is a resolving set for $T$ if every two distinct vertices in $T$ have different distances to some vertex in $S$. We show that the minimum size of a determining set for an order $n$ tournament (its determining number) is bounded by $\lfloor n/3 \rfloor$, while the minimum size of a resolving set for an order $n$ strong tournament (its metric dimension) is bounded by $\lfloor n/2 \rfloor$. Both bounds are optimal.

Oriented graphs generated by random points on a circle

2000 ◽  
Vol 37 (2) ◽  
pp. 534-539 ◽  
Author(s):  
Yoshiaki Itoh ◽  
Hiroshi Maehara ◽  
Norihide Tokushige
Keyword(s):  
Probability Distribution ◽  
Food Webs ◽  
Cascade Model ◽  
Generation Model ◽  
Random Generation ◽  
Oriented Graphs ◽  
Dominance Relations ◽  
Long Period ◽  
Regular Tournament

Extending the cascade model for food webs, we introduce a cyclic cascade model which is a random generation model of cyclic dominance relations. Put n species as n points Q1,Q2,…, Qn on a circle. If the counterclockwise way from Qi to Qj on the circle is shorter than the clockwise way, we say Qi dominates Qj. Consider a tournament whose dominance relations are generated from the points on a circle by this rule. We show that when we take n mutually independently distributed points on the circle, the probability of getting a regular tournament of order 2r+1 as the largest regular tournament is equal to (n/(2r+1))/2n-1. This probability distribution is for the number of existing species after a sufficiently long period, assuming a Lotka-Volterra cyclic cascade model.

scholarly journals The partition of a strong tournament

2005 ◽  
Vol 290 (2-3) ◽  
pp. 211-220 ◽  
Author(s):  
Hao Li ◽  
Jinlong Shu
Keyword(s):  
Strong Tournament

Oriented graphs generated by random points on a circle

2000 ◽  
Vol 37 (02) ◽  
pp. 534-539 ◽  
Author(s):  
Yoshiaki Itoh ◽  
Hiroshi Maehara ◽  
Norihide Tokushige
Keyword(s):  
Probability Distribution ◽  
Food Webs ◽  
Cascade Model ◽  
Generation Model ◽  
Random Generation ◽  
Oriented Graphs ◽  
Dominance Relations ◽  
Long Period ◽  
Regular Tournament

Extending the cascade model for food webs, we introduce a cyclic cascade model which is a random generation model of cyclic dominance relations. Put n species as n points Q 1,Q 2,…, Q n on a circle. If the counterclockwise way from Q i to Q j on the circle is shorter than the clockwise way, we say Q i dominates Q j . Consider a tournament whose dominance relations are generated from the points on a circle by this rule. We show that when we take n mutually independently distributed points on the circle, the probability of getting a regular tournament of order 2r+1 as the largest regular tournament is equal to (n/(2r+1))/2 n-1. This probability distribution is for the number of existing species after a sufficiently long period, assuming a Lotka-Volterra cyclic cascade model.

scholarly journals A result on tournaments with an application to counting

1977 ◽  
Vol 16 (2) ◽  
pp. 273-278 ◽  
Author(s):  
W.J.R. Eplett
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
First Passage ◽  
Number Of Solutions ◽  
Quadratic Residues ◽  
Regular Tournament

A generating function is derived for the number of transitions to the first passage through a particular vertex of a random walk on a doubly regular tournament. As an application of this result, we obtain a generating function for the number of solutions (q1, …, qk) of qi ≡ r (mod p), where p is a prime of the form 4l + 3 and the qi are quadratic residues modulo p.

scholarly journals A Note on the Number of Hamiltonian Paths in Strong Tournaments

10.37236/1141 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Arthur H. Busch
Keyword(s):  
Hamiltonian Paths ◽  
Minimum Number ◽  
Strong Tournament

We prove that the minimum number of distinct hamiltonian paths in a strong tournament of order $n$ is $5^{{n-1}\over{3}}$. A known construction shows this number is best possible when $n \equiv 1 \hbox{ mod } 3$ and gives similar minimal values for $n$ congruent to $0$ and $2$ modulo $3$.

scholarly journals Doubly regular tournaments of Szekeres type

1982 ◽  
Vol 32 (3) ◽  
pp. 399-404 ◽  
Author(s):  
Noboru Ito
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Hadamard Matrix ◽  
Quadratic Residue ◽  
Residue Type ◽  
Regular Tournament

AbstractThe purpose of this note is to determine the automorphism group of the doubly regular tournament of Szekeres type, and to use it to show that the corresponding skew Hadamard matrix H of order 2(q + 1), where q ≡5(mod 8) and q > 5, is not equivalent to the skew Hadamard matrix H(2q + 1) of quadratic residue type when 2q + 1 is a prime power.

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