Tree Embeddings and Tree-Star Ramsey Numbers
Zilong Yan, Yuejian Peng

TL;DR
This paper investigates conditions under which large trees can be embedded into graphs with high minimum degree, confirming a conjecture with a minor exception and applying findings to Ramsey numbers involving trees and stars.
Contribution
It provides a nearly complete characterization for embedding large trees with maximum degree at most n-3 into graphs with minimum degree at least n-3, refining a prior conjecture.
Findings
Confirmed Guo and Volkmann's conjecture with one exception.
Established necessary and sufficient conditions for embedding trees with maximum degree n-3.
Applied results to determine Ramsey numbers for trees versus stars.
Abstract
We say that a graph can be embedded into a graph if contains an isomorphic copy of as a subgraph. Guo and Volkmann \cite{GV} conjectured that if is a connected graph with at least vertices and minimum degree at least , then any tree with vertices and maximum degree at most can be embedded into . In this paper, we give a result slightly stronger than this conjecture and obtain a sufficient and necessary condition that a tree with vertices and maximum degree at most can be embedded into a connected graph G with at least vertices and minimum degree at least . Our result implies that the conjecture of Guo and Volkmann is true with one exception. We also give an application to the Ramsey number of a tree versus a star.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
