the colorful paths and rainbow paths have been considered by severalauthors.a colorful directed path in a digraph $g$ is a directed path with $chi(g)$ vertices whose colors are different. a $v$-colorful directed path is such a directed path, starting from $v$. we prove that for a given $3$-regular triangle-free digraph $g$ determining whether there is a proper $chi(g)$-coloring of $g$such that ...

dynamic analysis and study of atomic force microscope in liquid environment is the main goal of this research. hydrodynamic and squeeze forces act on cantilever of atomic force microscope which works in liquid environment, so. in this paper the effect of different environmental and physical factors had studied on frequency response diagrams. the importance of frequency response analysis is stud...

A simplex spanned by a colored point set in Euclidean d-space is colorful if all vertices have distinct colors. The union of all full-dimensional colorful simplices spanned by a colored point set is called the colorful union. We show that for every d ∈ N, the maximum combinatorial complexity of the colorful union of n colored points in R is between Ω(n(d−1) 2 ) and O(n(d−1) 2 log n). For d = 2,...

The NP-hard Colorful Components problem is a graph partitioning problem on vertex-colored graphs. We identify a new application of Colorful Components in the correction of Wikipedia interlanguage links, and describe and compare three exact and two heuristic approaches. In particular, we devise two ILP formulations, one based on Hitting Set and one based on Clique Partition. Furthermore, we use ...

The colorful Carathéodory theorem, proved by Bárány in 1982, states that given d+1 sets of points S1, . . . ,Sd+1 in R , such that each Si contains 0 in its convex hull, there exists a set T ⊆ ⋃ d+1 i=1 Si containing 0 in its convex hull and such that |T ∩Si| ≤ 1 for all i ∈ {1, . . . , d + 1}. An intriguing question – still open – is whether such a set T , whose existence is ensured, can be fo...