41 research outputs found
Domination in transitive colorings of tournaments
An edge coloring of a tournament T with colors 1,2,…,k is called \it k-transitive \rm 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 Erd\H os, Sands, Sauer and Woodrow (so the conjecture is interesting even if false). We show that the conjecture implies a stronger conjecture, a possible extension of a result of B\'ar\'any 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
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments
A digraph such that every proper induced subdigraph has a kernel is said to
be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI
for short) resp.) if the digraph has a kernel (does not have a kernel resp.).
The unique CKI-tournament is and the unique
KP-tournaments are the transitive tournaments, however bipartite tournaments
are KP. In this paper we characterize the CKI- and KP-digraphs for the
following families of digraphs: locally in-/out-semicomplete, asymmetric
arc-locally in-/out-semicomplete, asymmetric -quasi-transitive and
asymmetric -anti-quasi-transitive -free and we state that the problem
of determining whether a digraph of one of these families is CKI is polynomial,
giving a solution to a problem closely related to the following conjecture
posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for
locally in-semicomplete digraphs.Comment: 13 pages and 5 figure
The weakness of the Erd\H{o}s-Moser theorem under arithmetic reductions
The Erd\H{o}s-Moser theorem says that every infinite
tournament admits an infinite transitive subtournament. We study the
computational behavior of the Erd\H{o}s-Moser theorem with respect to the
arithmetic hierarchy, and prove that instances of
admit low solutions for every , and that if a set is
not arithmetical, then every instance of admits a solution
relative to which is still not arithmetical. We also provide a level-wise
refinement of this theorem. These results are part of a larger program of
computational study of combinatorial theorems in Reverse Mathematics