This paper is devoted to the development of algorithms for finding unit distance graphs with chromatic number greater than 4, embedded in a two-dimensional sphere or plane. Such graphs provide a lower bound for the Nelson-Hadwiger problem on the chromatic number of the plane and its generalizations to the case of the sphere. A series of 5-chromatic unit distance graphs on 64513 vertices embedded into the plane is constructed. Unlike previously known examples, these graphs do not contain the Moser spindle as a subgraph. The construction of 5-chromatic graphs embedded in a sphere at two values of the radius is given. Namely, the 5-chromatic unit distance graph on 372 vertices embedded into the circumsphere of an icosahedron with a unit edge length, and the 5-chromatic graph on 972 vertices embedded into the circumsphere of a great icosahedron are constructed.

No abstract

В работе изучается задача нахождения наименьшего числа переменных формулы первого порядка, записывающей свойство input-графа содержать подграф, изоморфный заданному pattern-графу. Рассматриваются input-графы, имеющие достаточно большую связность. Ранее эта задача была решена для всех pattern-графов с 4 вершинами, кроме двух - простого цикла и diamond-графа. В настоящей работе мы нашли значение этой величины для двух оставшихся графов. Библиография: 8 названий.

We propose a novel approach to the problem of mutual information (MI) estimation via introducing normalizing flows-based estimator. The estimator maps original data to the target distribution with known closed-form expression for MI. We demonstrate that our approach yields MI estimates for the original data. Experiments with high-dimensional data are provided to show the advantages of the proposed estimator.