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Bodlaender & Fomin (2005) showed that, given a graph G and a number c of colors, it is possible to test whether G admits an equitable c-coloring in time O(n O(t)), where t is the treewidth of G; in particular, equitable coloring may be solved optimally in polynomial time for trees (previously known due to Chen & Lih 1994) and outerplanar graphs.
Example of a National Park Passport Stamp for the Boston Harbor Islands National Recreation Area. Passport to Your National Parks is a program through which ink stamps can be acquired at no cost at park visitor centers and ranger stations at nearly all of the 433 units of the United States National Park System and most of the National Park Service's affiliated areas.
The modern passports were issued in 2012 BS. The first of this kind was issued in 10 Shrawan 2012 BS which was issued by the secretary of the ministry. These passports were in a form of sheet paper. The booklet form of the passport was issued in the 1960s. It is estimated that about 1 lakh passport was issued in sheet paper form. [2]
In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edges so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face (or ...
However, like the standard 3-edge-coloring problem, finding a coloring of this type is NP-complete. Total coloring is a form of coloring that combines vertex and edge coloring, by requiring both the vertices and edges to be colored. Any incident pair of a vertex and an edge, or an edge and an edge, must have distinct colors, as must any two ...
Quadratic programming (NP-hard in some cases, P if convex) Subset sum problem [3]: SP13 Variations on the Traveling salesman problem. The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric.
3-dimensional matchings. (a) Input T. (b)–(c) Solutions. In the mathematical discipline of graph theory, a 3-dimensional matching is a generalization of bipartite matching (also known as 2-dimensional matching) to 3-partite hypergraphs, which consist of hyperedges each of which contains 3 vertices (instead of edges containing 2 vertices in a usual graph).
103 is a prime number, and the largest prime factor of ! + = =. [1] The previous prime is 101.This makes 103 a twin prime. [2] It is the fifth irregular prime, [3] because it divides the numerator of the Bernoulli number = =.