Hadwiger Conjecture
There are several conjectures known as the Hadwiger conjecture or Hadwiger's conjecture. They include: * Hadwiger conjecture (graph theory), a relationship between the number of colors needed by a given graph and the size of its largest clique minor * Hadwiger conjecture (combinatorial geometry) that for any ''n''-dimensional convex body, at most 2''n'' smaller homothetic bodies are necessary to contain the original * Hadwiger's conjecture on dissection into orthoschemes See also * Hadwiger–Nelson problem on the chromatic number of unit distance graphs in the Euclidean plane * Hadwiger's theorem In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in \R^n. It was proved by Hugo Hadwiger. Introduction Valuations Let \mathbb^n be the collection of all c ...

Hadwiger Conjecture (graph Theory)
In graph theory, the Hadwiger conjecture states that if G is loopless and has no K_t minor then its chromatic number satisfies It is known to be true for The conjecture is a generalization of the four-color theorem and is considered to be one of the most important and challenging open problems in the field. In more detail, if all proper colorings of an undirected graph G use k or more colors, then one can find k disjoint connected subgraphs of G such that each subgraph is connected by an edge to each other subgraph. Contracting the edges within each of these subgraphs so that each subgraph collapses to a single vertex produces a complete graph K_k on k vertices as a minor This conjecture, a far-reaching generalization of the four-color problem, was made by Hugo Hadwiger in 1943 and is still unsolved. call it "one of the deepest unsolved problems in graph theory." Equivalent forms An equivalent form of the Hadwiger conjecture (the contrapositive of the form stated above ...
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Hadwiger Conjecture (combinatorial Geometry)
In combinatorial geometry, the Hadwiger conjecture states that any convex body in ''n''-dimensional Euclidean space can be covered by 2''n'' or fewer smaller bodies homothetic with the original body, and that furthermore, the upper bound of 2''n'' is necessary if and only if the body is a parallelepiped. There also exists an equivalent formulation in terms of the number of floodlights needed to illuminate the body. The Hadwiger conjecture is named after Hugo Hadwiger, who included it on a list of unsolved problems in 1957; it was, however, previously studied by and independently, . Additionally, there is a different Hadwiger conjecture concerning graph coloring—and in some sources the geometric Hadwiger conjecture is also called the Levi–Hadwiger conjecture or the Hadwiger–Levi covering problem. The conjecture remains unsolved even in three dimensions, though the two dimensional case was resolved by . Formal statement Formally, the Hadwiger conjecture is: If ''K'' is any ...
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Dissection Into Orthoschemes
In geometry, it is an unsolved conjecture of Hugo Hadwiger that every simplex can be dissected into orthoschemes, using a number of orthoschemes bounded by a function of the dimension of the simplex. If true, then more generally every convex polytope could be dissected into orthoschemes. Definitions and statement In this context, a simplex in d-dimensional Euclidean space is the convex hull of d+1 points that do not all lie in a common hyperplane. For example, a 2-dimensional simplex is just a triangle (the convex hull of three points in the plane) and a 3-dimensional simplex is a tetrahedron (the convex of four points in three-dimensional space). The points that form the simplex in this way are called its vertices. An orthoscheme, also called a path simplex, is a special kind of simplex. In it, the vertices can be connected by a path, such that every two edges in the path are at right angles to each other. A two-dimensional orthoscheme is a right triangle. A three-dimensional or ...
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Hadwiger–Nelson Problem
In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color. The answer is unknown, but has been narrowed down to one of the numbers 5, 6 or 7. The correct value may depend on the choice of axioms for set theory. Relation to finite graphs The question can be phrased in graph theoretic terms as follows. Let ''G'' be the unit distance graph of the plane: an infinite graph with all points of the plane as vertices and with an edge between two vertices if and only if the distance between the two points is 1. The Hadwiger–Nelson problem is to find the chromatic number of ''G''. As a consequence, the problem is often called "finding the chromatic number of the plane". By the de Bruijn–Erdős theorem, a result of , the problem is equivalent (under the assumption of the axiom of choice) to that of fin ...
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