Tiling $H$ in dense graphs

TL;DR

This paper determines the extremal structures for tiling an H-shaped tree in dense graphs, revealing a contrast with bipartite cases and refuting a previous conjecture related to the Erdős Matching Conjecture.

Contribution

It identifies the asymptotic extremal constructions for tiling an H-shaped tree in dense graphs, challenging existing conjectures and expanding understanding of graph tiling problems.

Findings

Extremal construction close to the complement of two cliques.

Contrasts with bipartite graph tiling cases.

Refutes Lang's conjecture on generalizing the Erdős Matching Conjecture.

Abstract

We determine asymptotically the two extremal constructions for the tiling problem of the $H$-shaped tree. In particular, the first extremal construction is close to the complement of two cliques, in contrast to previously studied bipartite graphs, where the first extremal construction is close to the complement of a single clique. This result refutes one of Lang's conjectures [arXiv:2308.12281], which seeks to generalize the Erd\H{o}s Matching Conjecture.