A note on degree conditions for Ramsey goodness of trees
Zhidan Luo, Yuejian Peng
TL;DR
The paper investigates degree conditions on a graph $G$ that ensure a Ramsey-type guarantee $G\rightarrow (T_n,K_m)$ for trees $T_n$ and cliques $K_m$, extending the path case $P_n$ to general trees. It determines key three-color Ramsey numbers $r(K_{1,\ell},T_n,K_m)$ for $\ell=t(n-1)+1$ and derives a minimum-degree bound $\delta(G)\ge N-t(n-1)-1$ that forces $G\rightarrow (T_n,K_m)$ in a specified range of orders $N$, thereby generalizing and strengthening previous results. The work also improves bounds when $T_n$ is not a star, leveraging known Ramsey numbers for $K_{1,\ell}$ versus $T_n$ to obtain tighter degree conditions and partial confirmations of the Aragão–Marciano–Mendonça conjecture across additional parameter regimes. Together, these results advance understanding of Ramsey goodness for trees under degree constraints and refine the landscape of related conjectures.
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{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 $N\geq (n- 1)(m- 1)+ 1$ vertices. If $δ(G)\geq N- \lceil n/2\rceil$, 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 $δ(G)$. We further improve the lower bound when $T_{n}\neq K_{1, n- 1}$, which partially confirms their conjecture.