Completing Partial Packings of Bipartite Graphs

TL;DR

This paper proves that for bipartite graphs, the minimal extension needed to complete partial packings into full designs grows slower than linearly with the number of vertices, resolving a long-standing conjecture.

Contribution

The paper establishes tight bounds for the growth of the extension function and confirms the conjecture that it is sublinear, settling a major open problem in graph design theory.

Findings

Proved that $f(n;H) = o(n)$ for bipartite graphs.

Established tight bounds for the growth of $f(n;H)$ as $n$ increases.

Resolved a long-standing conjecture by F"uredi and Lehel.

Abstract

Given a bipartite graph $H$ and an integer $n$, let $f(n;H)$ be the smallest integer such that, any set of edge disjoint copies of $H$ on $n$ vertices, can be extended to an $H$-design on at most $n+f(n;H)$ vertices. We establish tight bounds for the growth of $f(n;H)$ as $n \rightarrow \infty$. In particular, we prove the conjecture of F\"uredi and Lehel \cite{FuLe} that $f(n;H) = o(n)$. This settles a long-standing open problem.