## Abstract

Let F_{k} be the set of all graphs on k vertices. For a graph G, a k-decomposition is a set of induced subgraphs of G, each isomorphic to an element of F_{k}, such that each pair of vertices of G is in exactly one element of the set. It is a fundamental result of Wilson that for all n = |V (G)| sufficiently large, G has a k-decomposition if and only if G is k-divisible, namely k−1 divides n−1 and^{k}_{2} divides^{n}_{2} . Let v ∈ R^{|F}k^{|} be indexed by F_{k}. For a kdecomposition L of G, let ν_{v}(L) =^{P}_{F∈Fk} v_{F}d_{L,F} where d_{L,F} is the fraction of elements of L that are isomorphic to F. Let ν_{v}(G) = max_{L} ν_{v}(L) and ν_{v}(n) = min{ν_{v}(G): |V (G)| = n} The sequence ν_{v}(n) has a limit so let ν_{v} = lim_{n→∞} ν_{v}(n). Replacing kdecompositions with their fractional relaxations, one obtains the (polynomial time computable) fractional analogue ν_{v}^{∗}(G) and the corresponding fractional values ν_{v}^{∗}(n) and ν_{v}^{∗}. Our first main result is that for each v ∈ R^{|F}k| ν_{v} = ν_{v}^{∗} . Furthermore, there is a polynomial time algorithm that produces a decomposition L of a k-decomposable graph such that ν_{v}(L) ≥ ν_{v} − o_{n}(1). A similar result holds when F_{k} is the family of all tournaments on k vertices or when F_{k} is the family of all edge-colorings of K_{k}. We use these results to obtain new and improved bounds on several decomposition results. For example, we prove that every n-vertex tournament which is 3-divisible (namely n = 1, 3 mod 6) has a triangle decomposition in which the number of directed triangles is less than 0.0222n^{2}(1 + o(1)) and that every 5-decomposable n-vertex graph has a 5-decomposition in which the fraction of cycles of length 5 is o_{n}(1).

Original language | English |
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Pages | 1221-1238 |

Number of pages | 18 |

DOIs | |

State | Published - 2019 |

Event | 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United States Duration: 6 Jan 2019 → 9 Jan 2019 |

### Conference

Conference | 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 |
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Country/Territory | United States |

City | San Diego |

Period | 6/01/19 → 9/01/19 |

### Bibliographical note

Funding Information:This research was supported by the Israel Science Foundation (grant No. 1082/16).

Publisher Copyright:

Copyright © 2019 by SIAM

## ASJC Scopus subject areas

- Software
- General Mathematics