On the Ramsey number for paths and beaded wheels

Abstract

For given graphs GG and HH, the Ramsey number R(G,H)R(G,H) is the least natural number n such that for every graph FF of order nn the following condition holds: either FF contains GG or the complement of FF contains HH. Beaded wheel BW2,mBW2,m is a graph of order 2m+12m+1 which is obtained by inserting a new vertex in each spoke of the wheel WmWm. In this paper, we determine the Ramsey number of paths versus Beaded wheels: R(Pn,BW2,m)=2n−1R(Pn,BW2,m)=2n−1 or 2n2n if m≥3m≥3 is even or odd, respectively, provided n≥2m2−5m+4n≥2m2−5m+4.