  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.
