Completing Partial Packings of Bipartite Graphs
Zolt\'an F\"uredi, Ago-Erik Riet, Mykhaylo Tyomkyn

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 and an integer , let be the smallest integer such that, any set of edge disjoint copies of on vertices, can be extended to an -design on at most vertices. We establish tight bounds for the growth of as . In particular, we prove the conjecture of F\"uredi and Lehel \cite{FuLe} that . This settles a long-standing open problem.
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Taxonomy
TopicsLimits and Structures in Graph Theory · graph theory and CDMA systems · Advanced Graph Theory Research
