The extremal function for partial bipartite tilings

TL;DR

This paper determines the minimum edge density threshold needed in large graphs to ensure the presence of many disjoint copies of a fixed bipartite graph H, advancing understanding of extremal tiling problems.

Contribution

It establishes the exact threshold T_H(c) for partial bipartite tilings, extending previous extremal graph theory results with a novel technique based on Komlos's method.

Findings

Derived the threshold T_H(c) for partial bipartite tilings

Proved that graphs with density above T_H(c) contain nearly c/v(H) proportion of H copies

Introduced a variant of Komlos's technique for this problem

Abstract

For a fixed bipartite graph H and given number c, 0<c<1, we determine the threshold T_H(c) which guarantees that any n-vertex graph with at edge density at least T_H(c) contains $(1-o(1))c/v(H) n$ vertex-disjoint copies of H. In the proof we use a variant of a technique developed by Komlos~\bcolor{[Combinatorica 20 (2000), 203-218}]