On the Ramsey numbers of odd-linked double stars

The linked double star S-c(n, m), where n >= m >= 0, is the graph consisting of the union of two stars K-1,K-n and K-1,K-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 >= c, or if n <= left perpendicularc/2right perpendicular - 2 and m = 2. (C) 2022 The Author(s). Published by Elsevier B.V.