A note on degree conditions for Ramsey goodness of trees

Abstract

For given graphs $G_1,G_2$ and $G$, let $G\to (G_1,G_2)$ denote that each red-blue-coloring of $E(G)$ yields a red copy of $G_1$ or a blue copy of $G_2$. Arag{ã}o, Marciano and Mendon{\c c}a [L. Arag{ã}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 $N\ge (n-1)(m-1)+1$ vertices. If $\delta (G)\ge N-\lceil n/2\rceil$, then $G\to (P_n,K_m)$, 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 $T_n\ne K_{1,n-1}$, which partially confirms their conjecture.