The 2-dipath chromatic number of Halin graphs

A 2-dipath k-coloring f of an oriented graph is a mapping from to the color set {1,2,…,k} such that f(x)≠f(y) whenever two vertices x and y are linked by a directed path of length 1 or 2. The 2-dipath chromatic number of is the smallest k such that has a 2-dipath k-coloring. In this paper we prove that if is an oriented Halin graph, then . There exist infinitely many oriented Halin graphs such that .     

Minimum distance between a canal surface and a simple surface

The computation of the minimum distance between two objects is an important problem in the applications such as haptic rendering, CAD/CAM, NC verification, robotics and computer graphics. This paper presents a method to compute the minimum distance between a canal surface and a simple surface (i.e. a plane, a natural quadric, or a torus) by finding roots of a function of a single parameter. We utilize the fact that the normals at the closest points between two surfaces are collinear. Given the spine curve C(t), t_min≤t≤t_max, and the radius function r(t) for a canal surface, a point on the spine curve uniquely determines a characteristic circle on the surface. Normals to the canal surface at points on form a cone with a vertex and an axis which is parallel to Then we construct a function of t which expresses the condition that the perpendicular from C(t) to a given simple surface is embedded in the cone of normals to the canal surface at points on K(t). By solving this equation, we find characteristic circles which contain the points of locally minimum distance from the simple surface. Based on these circles, we can compute the minimum distance between given surfaces.