In this article, we study the so-called rectifying curves in an arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. If this fixed point is chosen to be the origin, then this condition is equivalent to saying that the position vector of the curve in every point lies in the orthogonal complement of its normal vector. Here we characterize rectifying curves in the n -dimensional Euclidean space in different ways: using conditions on their curvatures, with an expression for the tangential component, the normal component, or the binormal components of their position vector, and by constructing them starting from an arclength parameterized curve on the unit hypersphere.
List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory.Given a k-list-assignment L of a graph G, which is the assignment of a list L(v) of k colours to each vertex v ∈ V (G), we study the existence of k pairwise-disjoint proper colourings of G using colours from these lists. We may refer to this as a list-packing. Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest k for which such a list-packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of G. (The reader might already find it interesting that such a minimal k is well defined.) We also pursue a more focused study of the case when G is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal k above is not too much larger than the list chromatic number.Our study has taken inspiration from study of the strong chromatic number, and we also explore generalisations of the problem above in the same spirit.
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