Packing subdivisions into regular graphs
Richard Montgomery, Kalina Petrova, Arjun Ranganathan, Jane Tan

TL;DR
This paper proves that large regular graphs can be almost entirely covered by vertex-disjoint subdivisions of any fixed graph, confirming a long-standing conjecture and improving previous bounds on the degree requirement.
Contribution
It establishes that for any fixed graph and small error margin, large regular graphs can be nearly fully covered by subdivisions, confirming Verstra"ete's 2002 conjecture.
Findings
Verifies Verstra"ete's conjecture from 2002.
Shows coverage of at least (1-η)n vertices by subdivisions.
Reduces degree requirement from polylogarithmic to a fixed threshold.
Abstract
We show that, for any graph and , there exists a such that every -vertex -regular graph with has a collection of vertex-disjoint -subdivisions covering at least vertices. This verifies a conjecture of Verstra\"ete from 2002 and improves a recent result of Letzter, Methuku and Sudakov which additionally required to be at least polylogarithmic in .
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Taxonomy
TopicsFinite Group Theory Research · Advanced Graph Theory Research · Graph Labeling and Dimension Problems
