Tournaments with kernels by monochromatic paths

In this paper we prove the existence of kernels by monochromatic paths in m-coloured tournaments in which every cyclic tournament of order 3 is atmost 2-coloured in addition to other restrictions on the colouring ofcertain subdigraphs. We point out that in all previous results on kernelsby monochromatic paths in arc coloured tournaments, certain smallsubstructures are required to be monochromatic or monochromatic with atmost one exception, whereas here we allow up to three colours in two smallsubstructures

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

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