The study of classical Ramsey numbers R(mn) shows little
progress in the last two decades. Only nine classical Ramsey
numbers are known. This diculty ofnding the classical Ramsey
numbers has inspired many people to study generalizations of
classical Ramsey number. One of them is to determine Ramsey
number R(GH) for general graphs G and H (not necessarily
complete).

One of the most general results on graph Ramsey numbers is the
establishment of a general lower bound by Chvatal and Harary
[17] which is formulated as: R(GH) ((H) 1)(c(G) 1) + 1, where G
is a graph having no isolated vertices,

(H) is the chromatic number of H and c(G) denotes the
cardinality of large connected component of G.

Recently, Surahmat and Tomescu [41] studied the Ramsey number of
a combination of path Pn versus Jahangir graph J2m. They proved
that R(Pn J2m) = n+m1 for m 3 and n (4m 1)(m 1) + 1.
Furthermore, they determined that R(P4J22) = 6 and R(Pn J22) = n
+ 1 for n 5.

This dissertation studies the determination of Ramsey number for
a combination of path Pn and a wheel-like graph. What we mean by
wheel-like graph, is a graph obtained from a wheel by a graph
operation such as deletion or subdivision of the spoke edges.
The classes of wheel-like graphs which we consider are Jahangir
graph, generalized Jahangir graph and beaded wheel. First of all
we evaluate the Ramsey number for path Pn with respect to
Jahangir graph J2m. We improve the result of Surahmat and
Tomescu for m = 345 with n 2m + 1. Also, we determine the Ramsey
number for disjoint union of k identical copies of path Pn
versus Jahangir

graph J2m for m 2.

Moreover, we determine
the Ramsey number of path Pn versus generalized Jahangir graph Jsm
for dierent values of sm and n. We also, evaluate the Ramsey number
for combination of disjoint union of t identical copies of path
versus generalized Jahangir graph Jsm for even s 2 and m 3. At the
end, we nd the Ramsey number of path versus beaded wheel BW2m, i.e.
R(Pn BW2m) = 2n 1 or 2n if m 3 is even or odd, respectively,
provided n 2m2 5m + 4.