On the Ramsey Numbers of Odd-Linked Double Stars

The linked double star \(S_c(n,m)\), where \(n \geq m \geq 0\), is the graph consisting of the union of two stars \(K_{1,n}\) and \(K_{1,m}\) with a path on \(c\) vertices joining the centers. Its ramsey number \(r(S_c(n,m))\) is the smallest integer \(r\) such that every \(2\)-coloring of the edges of a \(K_r\) admits a monochromatic \(S_c(n,m)\). In this paper, we study the ramsey numbers of linked double stars when \(c\) is odd. In particular, we establish bounds on the value of \(r(S_c(n,m))\) and determine the exact value of \(r(S_c(n,m))\) if \(n \geq c\), or if \(n \leq \lfloor \frac{c}{2} \rfloor - 2\) and \(m = 2\).