Large dominating sets in tournamentsIs the feedback vertex number bounded by the maximum number of leaves in a spanning tree?What is the best lower bound for the domination number in regular graphs of girth 5?Rock-paper-scissors…expected number of cycles in a “random” bipartite directed graphProperties of a smallest tournament with domination number $k$Maximum number of hyperedges on a hypergraph without a spanning treefinding dominating cycles in $2K_2$-free graphsHamming graph and independent setsDoes every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $Omega(n)$?Minimum dominating sets in tournaments
Large dominating sets in tournaments
Is the feedback vertex number bounded by the maximum number of leaves in a spanning tree?What is the best lower bound for the domination number in regular graphs of girth 5?Rock-paper-scissors…expected number of cycles in a “random” bipartite directed graphProperties of a smallest tournament with domination number $k$Maximum number of hyperedges on a hypergraph without a spanning treefinding dominating cycles in $2K_2$-free graphsHamming graph and independent setsDoes every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $Omega(n)$?Minimum dominating sets in tournaments
$begingroup$
It is known that in any tournament with $n$ vertices, there is a dominating set of size no more than $lceil log_2 nrceil$. (See Fact 2.5 here.)
What are tournaments such that any dominating set is of size $Omega(log n)$? No example is given in the link above. A tournament that does not work is one where the vertices are on a cycle and each vertex has an edge to $(n-1)/2$ following vertices clockwise -- in this case taking two opposite vertices already gives a dominating set.
co.combinatorics graph-theory
asked May 16 at 17:47
pi66pi66
284110
$endgroup$
add a comment |
$begingroup$
It is known that in any tournament with $n$ vertices, there is a dominating set of size no more than $lceil log_2 nrceil$. (See Fact 2.5 here.)
What are tournaments such that any dominating set is of size $Omega(log n)$? No example is given in the link above. A tournament that does not work is one where the vertices are on a cycle and each vertex has an edge to $(n-1)/2$ following vertices clockwise -- in this case taking two opposite vertices already gives a dominating set.
co.combinatorics graph-theory
asked May 16 at 17:47
pi66pi66
284110
$endgroup$
add a comment |
$begingroup$
It is known that in any tournament with $n$ vertices, there is a dominating set of size no more than $lceil log_2 nrceil$. (See Fact 2.5 here.)
What are tournaments such that any dominating set is of size $Omega(log n)$? No example is given in the link above. A tournament that does not work is one where the vertices are on a cycle and each vertex has an edge to $(n-1)/2$ following vertices clockwise -- in this case taking two opposite vertices already gives a dominating set.
co.combinatorics graph-theory
asked May 16 at 17:47
pi66pi66
284110
$endgroup$
It is known that in any tournament with $n$ vertices, there is a dominating set of size no more than $lceil log_2 nrceil$. (See Fact 2.5 here.)
What are tournaments such that any dominating set is of size $Omega(log n)$? No example is given in the link above. A tournament that does not work is one where the vertices are on a cycle and each vertex has an edge to $(n-1)/2$ following vertices clockwise -- in this case taking two opposite vertices already gives a dominating set.
co.combinatorics graph-theory
co.combinatorics graph-theory
asked May 16 at 17:47
pi66pi66
284110
asked May 16 at 17:47
pi66pi66
284110
asked May 16 at 17:47
pi66pi66
284110
asked May 16 at 17:47
pi66pi66
284110
asked May 16 at 17:47
pi66pi66
284110
284110
add a comment |
add a comment |
1 Answer
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$begingroup$
All logarithms are base-$2$ here.
With high probability, a random tournament has no dominating sets of size $kapproxlog n-2loglog n$, as shown by Erdős [1].
Paley tournaments: if $qequiv3pmod4$ is a prime power, define a tournament whose vertex set is the finite field $mathbb F_q$ by
$$xto yiff y-xtext is a square in mathbb F_q.$$
Graham and Spencer [2] proved that the Paley tournament has no dominating sets of size $kapproxfrac12log q-loglog q$, as a consequence of Weil’s bound on character sums.Blass and Rossman [3] gave a somewhat complicated, but explicit construction of a tournament with an elementary proof that it has no dominating sets of size $kapprox(log n)^1/4$.
In all these examples, the tournaments actually have stronger properties: they satisfy the extension axioms up to size $k$ (i.e., whenever you select a set of $k$ vertices, and prescribe for each of them individually whether it wins or loses, there exists a vertex that fits these constraints).
References:
[1] Paul Erdős: On a problem in graph theory, Mathematical Gazette 47 (1963), no. 361, pp. 220-223.
[2] Ronald L. Graham, Joel H. Spencer: A constructive solution to a tournament problem, Canadian Mathematical Bulletin 14 (1971), no. 1, pp. 45-48.
[3] Andreas Blass, Benjamin Rossman: Explicit graphs with extension properties, Bulletin of the EATCS 86 (2005), pp. 166–175.
answered May 16 at 18:43
Emil JeřábekEmil Jeřábek
30.9k390144
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$begingroup$
All logarithms are base-$2$ here.
With high probability, a random tournament has no dominating sets of size $kapproxlog n-2loglog n$, as shown by Erdős [1].
Paley tournaments: if $qequiv3pmod4$ is a prime power, define a tournament whose vertex set is the finite field $mathbb F_q$ by
$$xto yiff y-xtext is a square in mathbb F_q.$$
Graham and Spencer [2] proved that the Paley tournament has no dominating sets of size $kapproxfrac12log q-loglog q$, as a consequence of Weil’s bound on character sums.Blass and Rossman [3] gave a somewhat complicated, but explicit construction of a tournament with an elementary proof that it has no dominating sets of size $kapprox(log n)^1/4$.
In all these examples, the tournaments actually have stronger properties: they satisfy the extension axioms up to size $k$ (i.e., whenever you select a set of $k$ vertices, and prescribe for each of them individually whether it wins or loses, there exists a vertex that fits these constraints).
References:
[1] Paul Erdős: On a problem in graph theory, Mathematical Gazette 47 (1963), no. 361, pp. 220-223.
[2] Ronald L. Graham, Joel H. Spencer: A constructive solution to a tournament problem, Canadian Mathematical Bulletin 14 (1971), no. 1, pp. 45-48.
[3] Andreas Blass, Benjamin Rossman: Explicit graphs with extension properties, Bulletin of the EATCS 86 (2005), pp. 166–175.
answered May 16 at 18:43
Emil JeřábekEmil Jeřábek
30.9k390144
$endgroup$
add a comment |
$begingroup$
All logarithms are base-$2$ here.
With high probability, a random tournament has no dominating sets of size $kapproxlog n-2loglog n$, as shown by Erdős [1].
Paley tournaments: if $qequiv3pmod4$ is a prime power, define a tournament whose vertex set is the finite field $mathbb F_q$ by
$$xto yiff y-xtext is a square in mathbb F_q.$$
Graham and Spencer [2] proved that the Paley tournament has no dominating sets of size $kapproxfrac12log q-loglog q$, as a consequence of Weil’s bound on character sums.Blass and Rossman [3] gave a somewhat complicated, but explicit construction of a tournament with an elementary proof that it has no dominating sets of size $kapprox(log n)^1/4$.
In all these examples, the tournaments actually have stronger properties: they satisfy the extension axioms up to size $k$ (i.e., whenever you select a set of $k$ vertices, and prescribe for each of them individually whether it wins or loses, there exists a vertex that fits these constraints).
References:
[1] Paul Erdős: On a problem in graph theory, Mathematical Gazette 47 (1963), no. 361, pp. 220-223.
[2] Ronald L. Graham, Joel H. Spencer: A constructive solution to a tournament problem, Canadian Mathematical Bulletin 14 (1971), no. 1, pp. 45-48.
[3] Andreas Blass, Benjamin Rossman: Explicit graphs with extension properties, Bulletin of the EATCS 86 (2005), pp. 166–175.
answered May 16 at 18:43
Emil JeřábekEmil Jeřábek
30.9k390144
$endgroup$
add a comment |
$begingroup$
All logarithms are base-$2$ here.
With high probability, a random tournament has no dominating sets of size $kapproxlog n-2loglog n$, as shown by Erdős [1].
Paley tournaments: if $qequiv3pmod4$ is a prime power, define a tournament whose vertex set is the finite field $mathbb F_q$ by
$$xto yiff y-xtext is a square in mathbb F_q.$$
Graham and Spencer [2] proved that the Paley tournament has no dominating sets of size $kapproxfrac12log q-loglog q$, as a consequence of Weil’s bound on character sums.Blass and Rossman [3] gave a somewhat complicated, but explicit construction of a tournament with an elementary proof that it has no dominating sets of size $kapprox(log n)^1/4$.
In all these examples, the tournaments actually have stronger properties: they satisfy the extension axioms up to size $k$ (i.e., whenever you select a set of $k$ vertices, and prescribe for each of them individually whether it wins or loses, there exists a vertex that fits these constraints).
References:
[1] Paul Erdős: On a problem in graph theory, Mathematical Gazette 47 (1963), no. 361, pp. 220-223.
[2] Ronald L. Graham, Joel H. Spencer: A constructive solution to a tournament problem, Canadian Mathematical Bulletin 14 (1971), no. 1, pp. 45-48.
[3] Andreas Blass, Benjamin Rossman: Explicit graphs with extension properties, Bulletin of the EATCS 86 (2005), pp. 166–175.
answered May 16 at 18:43
Emil JeřábekEmil Jeřábek
30.9k390144
$endgroup$
All logarithms are base-$2$ here.
With high probability, a random tournament has no dominating sets of size $kapproxlog n-2loglog n$, as shown by Erdős [1].
Paley tournaments: if $qequiv3pmod4$ is a prime power, define a tournament whose vertex set is the finite field $mathbb F_q$ by
$$xto yiff y-xtext is a square in mathbb F_q.$$
Graham and Spencer [2] proved that the Paley tournament has no dominating sets of size $kapproxfrac12log q-loglog q$, as a consequence of Weil’s bound on character sums.Blass and Rossman [3] gave a somewhat complicated, but explicit construction of a tournament with an elementary proof that it has no dominating sets of size $kapprox(log n)^1/4$.
In all these examples, the tournaments actually have stronger properties: they satisfy the extension axioms up to size $k$ (i.e., whenever you select a set of $k$ vertices, and prescribe for each of them individually whether it wins or loses, there exists a vertex that fits these constraints).
References:
[1] Paul Erdős: On a problem in graph theory, Mathematical Gazette 47 (1963), no. 361, pp. 220-223.
[2] Ronald L. Graham, Joel H. Spencer: A constructive solution to a tournament problem, Canadian Mathematical Bulletin 14 (1971), no. 1, pp. 45-48.
[3] Andreas Blass, Benjamin Rossman: Explicit graphs with extension properties, Bulletin of the EATCS 86 (2005), pp. 166–175.
answered May 16 at 18:43
Emil JeřábekEmil Jeřábek
30.9k390144
edited May 21 at 9:26
answered May 16 at 18:43
Emil JeřábekEmil Jeřábek
30.9k390144
answered May 16 at 18:43
Emil JeřábekEmil Jeřábek
30.9k390144
answered May 16 at 18:43
Emil JeřábekEmil Jeřábek
30.9k390144
30.9k390144
add a comment |
add a comment |
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