The Complexity of Perfect Matchings and Packings in Dense Hypergraphs

Given two k-graphs H and F , a perfect F -packing in H is a collection of vertexdisjoint 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 ≤ ` ≤ 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 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 `, k (i) resolves a conjecture of Keevash, Knox and Mycroft [20] whilst (ii) answers a question of Yuster [47] in the negative. 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.