2025/12/04 by Luo, Zhidan, Peng, Yuejian
Mathematics · Computer Science · #Limits and Structures in Graph Theory #Advanced Graph Theory Research #Advanced Topology and Set Theory
paper · doi:10.48550/arxiv.2512.04402
For given graphs G1, G2 and G, let G→ (G1, G2) denote that each red-blue-coloring of E(G) yields a red copy of G1 or a blue copy of G2. Aragão, Marciano and Mendon\c ca [L. Aragão, J. Pedro Marciano and W. Mendon\c ca, 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≥ (n- 1)(m- 1)+ 1 vertices. If δ(G)≥ N- \lceil n/2\rceil, then G→ (Pn, Km), where Pn is a tree on n vertices. In this note, we generalize Pn to any tree Tn with n vertices, and improve the lower bound of δ(G). We further improve the lower bound when Tn≠ K1, n- 1, which partially confirms their conjecture.