## Abstract

We prove that for every ∈ > 0 and positive integer r, there exists Δ_{0} = Δ_{0}(∈) such that if Δ > Δ_{0} and n > n(Δ, ∈, r) then there exists a packing of K_{n} with [(n - 1)/Δ] graphs, each having maximum degree at most Δ and girth at least r, where at most ∈n^{2} edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G, f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G, f), ranging over all possible assignments. Now let α_{Δ} be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ + 1 ≥ α_{Δ} ≥ Δ(1 - o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K_{n}).

Original language | English |
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Pages (from-to) | 579-587 |

Number of pages | 9 |

Journal | Graphs and Combinatorics |

Volume | 17 |

Issue number | 3 |

DOIs | |

State | Published - 2001 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics