Path-fan Ramsey numbers Original Research Article

For two given graphs image and image, the Ramsey number image is the smallest positive integer image such that for every graph image on image vertices the following holds: either image contains image as a subgraph or the complement of image contains image as a subgraph. In this paper, we study the Ramsey numbers image, where image is a path on image vertices and image is the graph obtained from image disjoint triangles by identifying precisely one vertex of every triangle (image is the join of image and image). We determine the exact values of image for the following values of image and image: image and image; image and image; image and image; image and image or image with image or image; odd image and image with image or image with image. Moreover, we give nontrivial lower bounds and upper bounds for image for the other values of image and image.