Theoretical Backgrounds of Applied Discrete Mathematics

Multiplicative residue semigroups with planar Cayley graphs

O. V. Knyazev, D. V. Solomatin

Omsk State Pedagogical University, Omsk, Russia

Abstract: We study multiplicative residue semigroups that admit planar Cayley graphs. It is proved that the multiplicative semigroup of the residue ring $\mathbb{Z}_n$ admits a planar Cayley graph if and only if $n=4,6,8$ or $n$ is a prime number. Examples of minimal systems of generators of multiplicative residue semigroups with respect to some modules and their Cayley graphs are given, illustrating the obtained results.

Keywords: residue, multiplicative semigroup of residues, generating semigroup set, Cayley graph of semigroup, planar graph.

UDC: 512.532.3

DOI: 10.17223/20710410/66/4