Path–kipas Ramsey numbers Original Research Article

For two given graphs F and H, the Ramsey number image is the smallest positive integer p such that for every graph image on p vertices the following holds: either image contains F as a subgraph or the complement of image contains H as a subgraph. In this paper, we study the Ramsey numbers image, where image is a path on n vertices and image is the graph obtained from the join of image and image. We determine the exact values of image for the following values of n and m: image and image; image and (m is odd, image) or (m is even, image); image and image or image; image and image or image or image with image or image; odd image and image with image or image with image. Moreover, we give lower bounds and upper bounds for image for the other values of m and n.