Vector clique decompositions

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Let Fk 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 Fk, 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 andk2 dividesn2 . Let v ∈ R|Fk| be indexed by Fk. For a kdecomposition L of G, let νv(L) =PF∈Fk vFdL,F where dL,F is the fraction of elements of L that are isomorphic to F. Let νv(G) = maxL νv(L) and νv(n) = min{νv(G): |V (G)| = n} The sequence νv(n) has a limit so let νv = limn→∞ νv(n). Replacing kdecompositions with their fractional relaxations, one obtains the (polynomial time computable) fractional analogue νv(G) and the corresponding fractional values νv(n) and νv. Our first main result is that for each v ∈ R|Fk| νv = νv . Furthermore, there is a polynomial time algorithm that produces a decomposition L of a k-decomposable graph such that νv(L) ≥ νv − on(1). A similar result holds when Fk is the family of all tournaments on k vertices or when Fk is the family of all edge-colorings of Kk. 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 0.0222n2(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 on(1).

Original languageEnglish
Number of pages18
StatePublished - 2019
Event30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United States
Duration: 6 Jan 20199 Jan 2019


Conference30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019
Country/TerritoryUnited States
CitySan Diego

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


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