Domination in transitive colorings of tournaments

Dömötör Pálvölgyi, A. Gyárfás

Research output: Article

5 Citations (Scopus)

Abstract

An edge coloring of a tournament T with colors 1, 2, ..., k is called k- transitive if the digraph T(i) defined by the edges of color i is transitively oriented for each 1 ≤ i≤ k. We explore a conjecture of the second author: For each positive integer k there exists a (least) p(k) such that every k-transitive tournament has a dominating set of at most p(k) vertices. We show how this conjecture relates to other conjectures and results. For example, it is a special case of a well-known conjecture of Erdos, Sands, Sauer and Woodrow [14] (so the conjecture is interesting even if false). We show that the conjecture implies a stronger conjecture, a possible extension of a result of Bárány and Lehel on covering point sets by boxes. The principle used leads also to an upper bound O(22d-1dlogd) on the d-dimensional box-cover number that is better than all previous bounds, in a sense close to best possible. We also improve the best bound known in 3 dimensions from 314 to 64 and propose possible further improvements through finding the maximum domination number over parity tournaments.

Original languageEnglish
Pages (from-to)1-11
Number of pages11
JournalJournal of Combinatorial Theory. Series B
Volume107
Issue number1
DOIs
Publication statusPublished - 2014

Fingerprint

Tournament
Coloring
Domination
Colouring
Color
Sand
Edge Coloring
Domination number
Dominating Set
Erdös
Digraph
Point Sets
Parity
Covering
Cover
Upper bound
Imply
Integer

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science
  • Computational Theory and Mathematics

Cite this

Domination in transitive colorings of tournaments. / Pálvölgyi, Dömötör; Gyárfás, A.

In: Journal of Combinatorial Theory. Series B, Vol. 107, No. 1, 2014, p. 1-11.

Research output: Article

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