## Abstract

Let n > m ≥ 4 be positive integers. The edge framing number efr(C_{m},C_{n}) of C_{m} and C_{n} is defined as the minimum size of a graph every edge of which belongs to an induced C_{m} and an induced C_{n}. We show that efr(C_{m},C_{n}) = n + 4 if n = 2m - 4 and m ≥ 5, efr(C_{m},C_{n}) = n + 5 if n = 2m - 6 and m ≥ 7 and efr(C_{m}, C_{n}) = n + 6 if n = 2m - 8 (m ≥ 10) or m = n - 1 (where n ≥ 5 and n ∉ {6,8}) or m = n - 2 (n = 6 or n ≥ 9). It is also shown that efr(C_{m},C_{n}) ≥ n + 6 for n > m ≥ 4 with n ≠ 2m - 4 or 2m - 6 and (m,n) ≠ (5,7). Furthermore, for the cases n = 2m - 4 (m ≥ 5) and n = 2m - 6 (m ≥ 7) we show that C_{m} and C_{n} are uniquely edge framed.

Original language | English |
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Pages (from-to) | 257-270 |

Number of pages | 14 |

Journal | Australasian Journal of Combinatorics |

Volume | 15 |

Publication status | Published - 1997 |

Externally published | Yes |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics