On Ramsey numbers for paths versus wheels

For two given graphs F and H, the Ramsey number R(F,H) 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 R(Pn,Wm), where Pn is a path on n vertices and Wm 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 R(Pn,Wm) for the following values of n and m: n = 1, 2, 3 or 5 and m ≥ 3; n = 4 and m = 3, 4, 5 or 7; n ≥ 6 and (m is odd, 3 ≤ m ≤ 2n− 1) or (m is even, 4 ≤ m ≤ n + 1); odd n ≥ 7 and m = 2n− 2 or m = 2n or m ≥ (n− 3); odd n ≥ 9 and q · n− 2q + 1 ≤ m ≤ q · n− q + 2 with 3 ≤ q ≤ n − 5. Moreover, we give nontrivial lower bounds and upper bounds for R(Pn,Wm) for the other values of m and n.