On Ramsey numbers for paths versus wheels 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 G on p vertices the following holds: either G contains F as a subgraph or the complement of G 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 a cycle on m vertices by adding a new vertex and edges joining it to all the vertices of the cycle. We present the exact values of image for the following values of n and m: image or 5 and image; image and image or 7; image and (m is odd, image) or (m is even, image); odd image and image or image or image; odd image and image with image. Moreover, we give nontrivial lower bounds and upper bounds for image for the other values of m and n