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arXiv · 2607.17808

Breaking the Bollob\'as-Eldridge-Catlin Barrier for Bipartite Graphs

Peter Allen·Julia Böttcher·Jozef Skokan·Benny Sudakov
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Abstract

The celebrated Bollob\'as-Eldridge-Catlin packing conjecture states that every $n$-vertex graph $G$ with minimum degree at least $\big(1-\frac{1}{\Delta+1}\big) n$ contains every $n$-vertex graph $H$ of maximum degree at most $\Delta$. Despite considerable attention, the conjecture remains widely open. We show that for bipartite $H$ this threshold can be greatly improved: there is an absolute constant $c>0$ such that every $n$-vertex graph $G$ with minimum degree at least $ \big(1-c\frac{\log\Delta}{\Delta}\big)n $ contains every $n$-vertex bipartite graph $H$ of maximum degree at most $\Delta$, provided $\Delta$ is not too large compared to $n$. Moreover, we prove that this logarithmic improvement is best possible up to the value of the constant.

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Peter Allen, Julia Böttcher, Jozef Skokan, Benny Sudakov. 2026-07-20. Breaking the Bollob\'as-Eldridge-Catlin Barrier for Bipartite Graphs. https://arxiv.org/abs/2607.17808

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