Abstract
Given two k-graphs H and F, a perfect F-packing in H is a collection of vertex-disjoint copies of F in H which together cover all the vertices in H. In the case when F is a single edge, a perfect F-packing is simply a perfect matching. For a given fixed F, it is often the case that the decision problem whether an n-vertex k-graph H contains a perfect F-packing is NP-complete. Indeed, if k≥3, the corresponding problem for perfect matchings is NP-complete [17,7] whilst if k=2 the problem is NP-complete in the case when F has a component consisting of at least 3 vertices [14].In this paper we give a general tool which can be used to determine classes of (hyper)graphs for which the corresponding decision problem for perfect F-packings is polynomial time solvable. We then give three applications of this tool: (i) Given 1≤ℓ≤k−1, we give a minimum ℓ-degree condition for which it is polynomial time solvable to determine whether a k-graph satisfying this condition has a perfect matching; (ii) Given any graph F we give a minimum degree condition for which it is polynomial time solvable to determine whether a graph satisfying this condition has a perfect F-packing; (iii) We also prove a similar result for perfect K-packings in k-graphs where K is a k-partite k-graph.For a range of values of ℓ,k (i) resolves a conjecture of Keevash, Knox and Mycroft [20]; (ii) answers a question of Yuster [47] in the negative; whilst (iii) generalises a result of Keevash, Knox and Mycroft [20]. In many cases our results are best possible in the sense that lowering the minimum degree condition means that the corresponding decision problem becomes NP-complete.
Highlights
Given k ≥ 2, a k-uniform hypergraph consists of a vertex set V (H) and an edge set E(H) ⊆ V (H) k, where every edge is a k-element subset of V (H)
We give three applications of this tool: (i) Given 1 ≤ ≤ k−1, we give a minimum -degree condition for which it is polynomial time solvable to determine whether a k-graph satisfying this condition has a perfect matching; (ii) Given any graph F we give a minimum degree condition for which it is polynomial time solvable to determine whether a graph satisfying this condition has a perfect F -packing; (iii) We prove a similar result for perfect K-packings in k-graphs where K is a k-partite k-graph
Where C ∈ {3/2, 2, 5/2, 3} depends on the value of n and k. Such results give us classes of dense k-graphs for which we are certain to have a perfect matching. This raises the question of whether one can lower the minimum -degree condition in Conjecture 1.1 whilst still ensuring it is decidable in polynomial time whether such a k-graph H has a perfect matching: Let PM(k, δ) denote the problem of deciding whether there is a perfect matching in a given k-graph on n vertices with minimum
Summary
Given k ≥ 2, a k-uniform hypergraph (or k-graph) consists of a vertex set V (H) and an edge set E(H) ⊆. The decision problem whether a k-graph contains a perfect matching is famously NP-complete for k ≥ 3 (see [17,7]). Hell and Kirkpatrick [14] showed that the decision problem whether a graph G has a perfect F -packing is NP-complete precisely when F has a component consisting of at least 3 vertices. Roughly speaking, for any k-graph F , Theorem 3.1 yields a general class of k-graphs within which we do have a complete characterisation of those k-graphs that contain a perfect F -packing. Each of our applications convey an underlying theme: In each case, the class of (hyper)graphs H we consider are those satisfying some minimum degree condition which ensures an almost perfect matching or packing M (i.e. M covers all but a constant number of the vertices of H). In each application we show that we can detect the ‘last obstructions’ to having a perfect matching or packing efficiently
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