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.
Categories: math.CO