A multipartite Hajnal-Szemerédi theorem

Perfect Matchings, Tilings and Hamilton Cycles in Hypergraphs

This thesis contains problems in finding spanning subgraphs in graphs, such as, perfect matchings, tilings and Hamilton cycles. First, we consider the tiling problems in graphs, which are natural generalizations of the matching problems. We give new proofs of the multipartite Hajnal-Szemerédi Theorem for the tripartite and quadripartite cases. Second, we consider Hamilton cycles in hypergraphs....

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.

On multipartite Hajnal-Szemerédi theorems

Let G be a k-partite graph with n vertices in parts such that each vertex is adjacent to at least δ∗(G) vertices in each of the other parts. Magyar and Martin [20] proved that for k = 3, if δ∗(G) ≥ 2 3 n and n is sufficiently large, then G contains a K3-factor (a spanning subgraph consisting of n vertex-disjoint copies of K3) except that G is one particular graph. Martin and Szemerédi [21] prov...

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...

An Extension of the Hajnal-Szemerédi Theorem to Directed Graphs