Researches of semigroups with planar Cayley graphs: results and problems

The classical Cayley graphs for groups have long and thoroughly proven themselves in solving applied problems, finding application in various scientific fields, from cryptography to campanology. Recently, more and more attention has been paid to the researching the applied aspects of Cayley graphs of semigroups. In this paper, we survey a result obtained in investigations on semigroups with planar Cayley graphs and results on the study of planarity rank of a semigroup varieties. A number of unsolved problems are formulated, the solution of which finds application in the combinatorial theory of semigroups. Just as Cayley graphs of semigroups are used for constructing hash functions, the concluding part of the paper introduces the concept of the spectrum of planarity ranks for varieties of semigroups, with the help of which it becomes possible to hash semigroup varieties.