A degree sequence Hajnal--Szemer\'edi theorem

Abstract

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--Szemer\'edi theorem characterises the minimum degree that ensures a graph $G$ contains a perfect $K_r$-packing. Balogh, Kostochka and Treglown proposed a degree sequence version of the Hajnal--Szemer\'edi theorem which, if true, gives a strengthening of the Hajnal--Szemer\'edi theorem. In this paper we prove this conjecture asymptotically. Another fundamental result in the area is the Alon--Yuster theorem which gives a minimum degree condition that ensures a graph contains a perfect $H$-packing for an \emph{arbitrary} graph $H$. We give a wide-reaching generalisation of this result by answering another conjecture of Balogh, Kostochka and Treglown on the degree sequence of a graph that forces a perfect $H$-packing. We also prove a degree sequence result concerning perfect transitive tournament packings in directed graphs. The proofs blend together the regularity and absorbing methods.