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













5












$begingroup$


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.










share|cite|improve this question









$endgroup$
















    5












    $begingroup$


    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.










    share|cite|improve this question









    $endgroup$














      5












      5








      5





      $begingroup$


      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.










      share|cite|improve this question









      $endgroup$




      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






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      asked May 16 at 17:47









      pi66pi66

      284110




      284110




















          1 Answer
          1






          active

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          8












          $begingroup$

          All logarithms are base-$2$ here.



          1. With high probability, a random tournament has no dominating sets of size $kapproxlog n-2loglog n$, as shown by Erdős [1].


          2. 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.


          3. 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.






          share|cite|improve this answer











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            8












            $begingroup$

            All logarithms are base-$2$ here.



            1. With high probability, a random tournament has no dominating sets of size $kapproxlog n-2loglog n$, as shown by Erdős [1].


            2. 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.


            3. 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.






            share|cite|improve this answer











            $endgroup$

















              8












              $begingroup$

              All logarithms are base-$2$ here.



              1. With high probability, a random tournament has no dominating sets of size $kapproxlog n-2loglog n$, as shown by Erdős [1].


              2. 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.


              3. 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.






              share|cite|improve this answer











              $endgroup$















                8












                8








                8





                $begingroup$

                All logarithms are base-$2$ here.



                1. With high probability, a random tournament has no dominating sets of size $kapproxlog n-2loglog n$, as shown by Erdős [1].


                2. 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.


                3. 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.






                share|cite|improve this answer











                $endgroup$



                All logarithms are base-$2$ here.



                1. With high probability, a random tournament has no dominating sets of size $kapproxlog n-2loglog n$, as shown by Erdős [1].


                2. 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.


                3. 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.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited May 21 at 9:26

























                answered May 16 at 18:43









                Emil JeřábekEmil Jeřábek

                30.9k390144




                30.9k390144



























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