## Abstract

In this paper we study the length of the longest induced cycle in the unit circulant graph X_{n} = Cay(ℤ;ℤ_{n}*), where ℤ* is the group of units in ℤ_{n}. Using residues modulo the primes dividing n, we introduce a representation of the vertices that reduces the problem to a purely combinatorial question of comparing strings of symbols. This representation allows us to prove that the multiplicity of each prime dividing n, and even the value of each prime (if sufficiently large) has no effect on the length of the longest induced cycle in X_{n}. We also see that if n has r distinct prime divisors, X_{n} always contains an induced cycle of length 2^{r} + 2, improving the r In r lower bound of Berrezbeitia and Giudici. Moreover, we extend our results for X_{n} to conjunctions of complete k_{i}-partite graphs, where k_{i} need not be finite, and also to unit circulant graphs on any quotient of a Dedekind domain.

Original language | English (US) |
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Journal | Electronic Journal of Combinatorics |

Volume | 12 |

Issue number | 1 R |

DOIs | |

State | Published - Oct 13 2005 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics