Tree Embeddings and Tree-Star Ramsey Numbers

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 $F$ can be embedded into a graph $G$ if $G$ contains an isomorphic copy of $F$ as a subgraph. Guo and Volkmann \cite{GV} conjectured that if $G$ is a connected graph with at least $n$ vertices and minimum degree at least $n-3$, then any tree with $n$ vertices and maximum degree at most $n-4$ can be embedded into $G$. In this paper, we give a result slightly stronger than this conjecture and obtain a sufficient and necessary condition that a tree with $n$ vertices and maximum degree at most $n-3$ can be embedded into a connected graph G with at least $n$ vertices and minimum degree at least $n-3$. 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.