A Multipartite Hajnal-Szemer\'edi Theorem

TL;DR

This paper proves the multipartite Hajnal-Szemerédi theorem for large graphs, confirming the minimum degree threshold for perfect K_k-packings in multipartite graphs, building on hypergraph matching results and stability analysis.

Contribution

It establishes the exact multipartite Hajnal-Szemerédi theorem for large graphs, extending previous approximate results to exact thresholds.

Findings

Proves the exact minimum degree threshold for perfect K_k-packings in large multipartite graphs.

Uses stability analysis to handle extremal cases.

Confirms Fischer's conjecture for sufficiently large graphs.

Abstract

The celebrated Hajnal-Szemer\'edi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K_k-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 this paper, we apply a stability analysis to the extremal cases of this argument, thus showing that the exact conjecture holds for any sufficiently large graph.