The Nonexistence of Certain Moufang Polygons.
The nonexistence of certain projective planes of order 15.
The nonexistence of regular near octagons with parameters .
The nonnegative -matrix completion problem.
The NP-completeness of automorphic colorings
Given a graph G, an automorphic edge(vertex)-coloring of G is a proper edge(vertex)-coloring such that each automorphism of the graph preserves the coloring. The automorphic chromatic index (number) is the least integer k for which G admits an automorphic edge(vertex)-coloring with k colors. We show that it is NP-complete to determine the automorphic chromatic index and the automorphic chromatic number of an arbitrary graph.
The number of 0-1-2 increasing trees as two different evaluations of the Tutte polynomial of a complete graph.
The number of 1-factors in some polyhexes.
The Number Of Antichains Of Finite Power Sets
The number of antichains of finite power sets.
The number of archiral trees.
The number of binomial coefficients in residue classes modulo p and .
The number of certain -combinations of an -set.
The number of complete exceptional sequences for a Dynkin algebra
The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper determines the number of complete exceptional sequences for any Dynkin algebra. Since the complete exceptional sequences for a Dynkin algebra of Dynkin type Δ correspond bijectively to the maximal chains in the lattice of non-crossing partitions of type Δ, the calculations presented here may also be considered as a categorification of the corresponding result for non-crossing partitions.
The number of crossings in a regular drawing of the complete bipartite graph.
The number of edges in critical graphs.
The number of edges on generalizations of Paley graphs.
The number of eigenvalues greater than two in the Laplacian spectrum of a graph
The number of elements in the mutation class of a quiver of type .
The number of -matchings in almost every tree is a zero residue.
The number of graphs not containing as a minor.