On the Ramsey numbers for linear forest versus some graphs

For given graphs G and H; the Ramsey number R(G;H) is the leastnatural number n such that for every graph F of order n the followingcondition holds: either F contains G or the complement of F contains H.In this paper firstly, we determine Ramsey number for union of pathswith respect to sunflower graphs, For m ≥ 3, the sunflower graph SFmis a graph on 2m + 1 vertices obtained by taking a wheel Wm with hubx, an m-cycle v1,v2,...,vm, and additional m vertices w1,w2,...,wm, wherewi is joined by edges to vi; vi+1 for i= 1, 2,...,m, where i + 1 is takenmodulo m. The hub of Wm is also called the hub of SFm. Secondly; wedetermine Ramsey numbers for a combination of a party graph. At theend, we propose some open problems for further work in this area ofresearch. References1. K. Ali, E. T. Baskoro, A. Q. Baig, On the Ramsey numbers forcombination of linear forest versus cocktail party graphs, JCMCC(2009), to appear.2. E. T. Baskoro, Surahmat, The Ramsey numbers of path with respectto wheels, Discrete Math., 294 (2005), 275-277.3. S. A. Burr, P. ErdÄos and J. H. Spencer, Ramsey theorem formultiple copies of graphs, Trans. Amer. Math. Soc., 209 (1975), 87-89.4. S. A. Burr, On Ramsey numbers for large disjoint unions of graphs,Discrete Math., 70 (1988), 277-293.5. Y. J. Chen, Y. Q. Zhang and K.M. Zhang, The Ramsey numbers forpaths versus wheels, Discrete Math., 290 (2005), 85-87.6. R. J. Faudree, R. H. Schelp, M. Simonovits, On some Ramsey typeproblems connected with paths, cycles and trees, Ars Combinatoria,29A(1990), 97-106.7. S. P. Radziszowski, Small Ramsey numbers, Electron. J. Combin.,2009, #DS1.9, http://www.combinatorics.org/