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Spanning trees in graphs without large bipartite holes

Part of: Graph theory

Published online by Cambridge University Press:  14 November 2023

Jie Han ,
Jie Hu ,
Lidan Ping ,
Guanghui Wang ,
Yi Wang  and
Donglei Yang
Jie Han
Affiliation:
School of Mathematics and Statistics and Center for Applied Mathematics, Beijing Institute of Technology, Beijing, China
Jie Hu
Affiliation:
Center for Combinatorics and LPMC, Nankai University, Tianjin, China
Lidan Ping
Affiliation:
School of Mathematics, Shandong University, Jinan, China
Guanghui Wang
Affiliation:
School of Mathematics, Shandong University, Jinan, China
Yi Wang
Affiliation:
Data Science Institute, Shandong University, Jinan, China
Donglei Yang*
Affiliation:
School of Mathematics, Shandong University, Jinan, China
*
Corresponding author: Donglei Yang; Email: dlyang@sdu.edu.cn
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Abstract

We show that for any $\varepsilon \gt 0$ and $\Delta \in \mathbb{N}$, there exists $\alpha \gt 0$ such that for sufficiently large $n$, every $n$-vertex graph $G$ satisfying that $\delta (G)\geq \varepsilon n$ and $e(X, Y)\gt 0$ for every pair of disjoint vertex sets $X, Y\subseteq V(G)$ of size $\alpha n$ contains all spanning trees with maximum degree at most $\Delta$. This strengthens a result of Böttcher, Han, Kohayakawa, Montgomery, Parczyk, and Person.

Keywords

Bipartite holesspanning treesabsorbing method

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C05: Trees
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 33 , Issue 3 , May 2024 , pp. 270 - 285
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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Footnotes

Jie Hu: Supported by Natural Science Foundation of China (12131013, 12161141006). Guanghui Wang: Research supported by Natural Science Foundation of China (12231018) and Young Taishan Scholars probgram of Shandong Province (201909001). Donglei Yang: Supported by the China Post-doctoral Science Foundation (2021T140413), Natural Science Foundation of China (12101365) and Natural ScienceFoundation of Shandong Province (ZR2021QA029).

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