Towards a Weighted Version of the Hajnal-Szemerédi Theorem

A degree sequence Hajnal-Szemerédi theorem

We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem [12] characterises the minimum degree that ensures a graph G contains a perfect Kr-packing. Balogh, Kostochka and Treglown [4] proposed a degree sequence version of the Hajnal–Szemerédi theorem which, if true, gives a strength...

Quadripartite version of the Hajnal-Szemerédi theorem

Let G be a quadripartite graph with N vertices in each vertex class and each vertex is adjacent to at least (3/4)N vertices in each of the other classes. There exists an N0 such that, if N ≥ N0, then G contains a subgraph that consists of N vertex-disjoint copies of K4.

A multipartite Hajnal-Szemerédi theorem

The celebrated Hajnal-Szemerédi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect Kk-packing. Fischer’s conjecture states that the analogous result holds for all multipartite graphs except for those formed by a single construction. Recently, we deduced an approximate version of this conjecture from new results on perfect matchings in hypergraphs. In thi...

An Extension of the Hajnal-Szemerédi Theorem to Directed Graphs

Approximate multipartite version of the Hajnal-Szemerédi theorem

Let q be a positive integer, and G be a q-partite simple graph on qn vertices, with n vertices in each vertex class. Let δ = k k+1 + λ, where k = q + 4dlog qe and λ is a positive real number. If each vertex of G is adjacent to at least δn vertices in each of the other vertex classes, q is bounded and n is large enough, then G has a Kq-factor.