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

Kernel perfect and critical kernel imperfect digraphs structure

A kernel $N$ of a digraph $D$ is an independent set of vertices of $D$ such that for every $w \in V(D)-N$ there exists an arc from $w$ to $N$. If every induced subdigraph of $D$ has a kernel, $D$ is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If $F$ is a set of arcs of $D$, a semikernel modulo $F$, $S$ of $D$ is an independent set of vertices of $D$ such that for every $z \in V(D)- S$ for which there exists an $Sz-$arc of $D-F$, there also exists an $zS-$arc in $D$. In this talk some structural results concerning critical kernel imperfect and sufficient conditions for a digraph to be a critical kernel imperfect digraph are presented

Kernels in edge-coloured orientations of nearly complete graphs

AbstractWe call the digraph D an orientation of a graph G if D is obtained from G by the orientation of each edge of G in exactly one of the two possible directions. The digraph D is an m-coloured digraph if the arcs of D are coloured with m-colours.Let D be an m-coloured digraph. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike.A set N⊆V(D) is said to be a kernel by monochromatic paths if it satisfies the two following conditions: (i) for every pair of different vertices u,v∈N there is no monochromatic directed path between them and (ii) for every vertex x∈V(D)-N there is a vertex y∈N such that there is an xy-monochromatic directed path.In this paper we obtain sufficient conditions for an m-coloured orientation of a graph obtained from Kn by deletion of the arcs of K1,r (0⩽r⩽n-1) to have a kernel by monochromatic