Kernels by Monochromatic Directed Paths in 3-Colored Tournaments and Quasi-Tournaments
We call the tournament T an m-coloured tournament if the arcs of T are coloured with m colours. If v is a vertex of an m-coloured tournament T, we denote by ζ(v) the set of colours assigned to the arcs with v as endpoint, ζ − (v) the set of colours assigned to the arcs with v as final vertex, and ζ + (v) the set of colours assigned to the arcs with v as an initial vertex. In this paper, we prove that if T is a 3-coloured tournament which does not contain C3 such that: 1. for each v ∈ V(T) |ζ − (v) |≤ 2,
Domination in transitive colorings of tournaments
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 Erdős, 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 Barany and Lehel on covering point sets by boxes. The principle used leads also to an upper bound O ( 2 2 d − 1 d log d ) 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 3 14 to 64 and propose possible further improvements through finding the maximum domination number over parity tournaments.
Shape and surface texture of molten droplets deposited on cold surfaces
When a molten droplet impacts on a cold surface, the interaction between solidification and fluid dynamics can lead to different shapes and surface textures of deposited droplets. This article presents an experimental study on deposition of small droplets (39 μm diameter) of molten wax ink on solid surfaces. Our results show that both the final shape and the surface texture of deposited droplets are extremely sensitive to deposition conditions, which include initial temperature of droplets, type of substrate surface and distance between substrate and printhead. Different surface textures are observed and shown to depend on the thermal and flow histories of ink droplets during the process of impact. A test using crystalline wax reveals crystallization as one of the major factors in determining the final texture.