### Abstract

In 2-edge-colored graphs, we define an (s, t)-cycle to be a cycle of length s + t, in which s consecutive edges are in one color and the remaining t edges are in the other color. Here we investigate the existence of (s, t)-cycles, in a 2-edge-colored complete graph K^{c}_{n} on n vertices. In particular, in the first result we give a complete characterization for the existence of (s, t)-cycles in K^{c}_{n} with n relatively large with respect to max({s, t}). We also study cycles of length 4 for all possible values' of s and t. Then, we show that K^{c}_{n} contains an (s, t)-hamiltonian cycle unless it is isomorphic to a specified graph. This extends a result of A. Gyárfás [Journal of Graph Theory, 7 (1983), 131-135]. Finally, we give some sufficient conditions for the existence of (s, 1)-cycles, ∀ s ∈ {2, 3, ⋯, n - 2}.

Original language | English |
---|---|

Pages (from-to) | 153-162 |

Number of pages | 10 |

Journal | Journal of Graph Theory |

Volume | 21 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 1996 |

### ASJC Scopus subject areas

- Geometry and Topology

## Fingerprint Dive into the research topics of 'Cycles of Given Color Patterns'. Together they form a unique fingerprint.

## Cite this

*Journal of Graph Theory*,

*21*(2), 153-162. https://doi.org/10.1002/(SICI)1097-0118(199602)21:2<153::AID-JGT4>3.0.CO;2-Q