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A Mycielski graph M_k of order k is a triangle-free graph with chromatic number k having the smallest possible number of vertices. For example, triangle-free graphs with ...

Every position of every impartial game has a nim-value, making it equivalent to a nim-heap. To find the nim-value (also called the Sprague-Grundy number), take the mex of the ...

A formula relating the number of polyhedron vertices V, faces F, and polyhedron edges E of a simply connected (i.e., genus 0) polyhedron (or polygon). It was discovered ...

A polyomino-like object made by attaching squares joined either at sides or corners. Because neighboring squares can be in relation to one another as kings may move on a ...

A graph with projective plane crossing number equal to 0 may be said to be projective planar. Examples of projective planar graphs with graph crossing number >=2 include the ...

A generalization of simple majority voting in which a list of quotas {q_0,...,q_n} specifies, according to the number of votes, how many votes an alternative needs to win ...

The number of ways N(m,n) of finding a subrectangle with an m×n rectangle can be computed by counting the number of ways in which the upper right-hand corner can be selected ...

An extended rooted binary tree satisfying the following conditions: 1. Every node has two children, each colored either red or black. 2. Every tree leaf node is colored ...

The regulator of a number field K is a positive number associated with K. The regulator of an imaginary quadratic field is 1 and that of a real quadratic, imaginary cubic, or ...

A formula for the permanent of a matrix perm(a_(ij))=(-1)^nsum_(s subset= {1,...,n})(-1)^(|s|)product_(i=1)^nsum_(j in s)a_(ij), where the sum is over all subsets of ...