A note on degree conditions for Ramsey goodness of trees

TL;DR

This paper extends previous results on degree conditions for Ramsey properties from paths to general trees, improving bounds and partially confirming a conjecture about trees with specific structures.

Contribution

It generalizes degree conditions for Ramsey goodness from paths to all trees and refines bounds, especially for non-star trees, advancing understanding of graph Ramsey theory.

Findings

Generalized degree conditions from paths to all trees.

Improved lower bounds for the minimum degree condition.

Partially confirmed the conjecture for certain trees.

Abstract

For given graphs $G_{1}, G_{2}$ and $G$, let $G\rightarrow (G_{1}, G_{2})$ denote that each red-blue-coloring of $E(G)$ yields a red copy of $G_{1}$ or a blue copy of $G_{2}$. Arag{\~a}o, Marciano and Mendon{\c c}a [L. Arag{\~a}o, J. Pedro Marciano and W. Mendon{\c c}a, Degree conditions for Ramsey goodness of paths, {\it European Journal of Combinatorics}, {\bf 124} (2025), 104082] proved the following. Let $G$ be a graph on $N\geq (n- 1)(m- 1)+ 1$ vertices. If $\delta(G)\geq N- \lceil n/2\rceil$, then $G\rightarrow (P_{n}, K_{m})$, where $P_{n}$ is a tree on $n$ vertices. In this note, we generalize $P_{n}$ to any tree $T_{n}$ with $n$ vertices, and improve the lower bound of $\delta(G)$. We further improve the lower bound when $T_{n}\neq K_{1, n- 1}$, which partially confirms their conjecture.