The Ramsey numbers \(R(T_n,W_6)\) for \(T_n\) without certain deletable sets (Q2583106)

Let \(W_m\) denote the wheel on \(m+1\) vertices, \(S_n\) a star on \(n\) vertices and \(T_n\) a tree with \(n\) vertices. In 2002, \textit{E. T. Baskoro, Surahmat, S. M. Nababan} and \textit{M. Miller} [Graphs Comb.~18, No. 4, 717--721 (2002; Zbl 1009.05098)] conjectured that if \(T_n\neq S_n\) and \(n+1\geq m \geq 4\), then \(R(T_n,W_m)=2n-1\) when \(m\) is even and \(R(T_n,W_m)=3n-2\) when \(m\) is odd, and they proved their conjecture for \(m=4,5\). The current authors showed [Appl.~Math.~Lett.~17, 281--285 (2004; Zbl 1055.05104)] that \(R(T_n,W_6)=2n\) when \(T_n\) is either \(S_{n-1}\) with one edge subdivided, or \(S_{n-2}\) with one edge subdivided twice (when \(n\) is divisible by 3), and conjectured that these are the only exceptions to the general conjecture when \(m=6\). They verified this in the cases when \(T_n\) is \(S_{n-2}\) with two edges subdivided or a double star with a vertex of degree 3, \(D_n\). In this paper the authors verify their conjecture when \(T_n\) is \(D_{n-1}\) with the edge connecting the high degree vertices subdivided once, when \(T_n\) is \(S_{n-3}\) with one edge subdivided 3 times, or one of a list of about 20 star-like trees on at most 11 vertices. The proofs consist mainly of a brief, but somewhat tedious, case analysis.