Gallai-Ramsey numbers of books

For a graph G, the Ramsey number R-k(G) is defined as the minimum rsuch that any k-edge coloring of K-r contains a monochromatic G. A Gallai k-coloring is a k-edge coloring of a complete graph without rainbow triangle, where a rainbow triangle is that all three edges are colored by three colors. The Gallai-Ramsey number GR(k)(G) is defined as the minimum rsuch that any Gallai k-coloring of K-r contains a monochromatic G. Call B-n book that is formed by ntriangles sharing a common edge. For all k >= 2 and n >= 1, we show that GR(k)(B-n) is at most 1.1(4n + 1)5(k-2/2) If k is even and 2.75(4n + 1)5(k-3/2) if k is odd, respectively. (c) 2024 Elsevier B.V. All rights reserved.