Abstract:
The study of classical Ramsey numbers R(m, n) shows little progress in the last two
decades. Only nine classical Ramsey numbers are known. This difficulty of finding
the classical Ramsey numbers has inspired many people to study generalizations of
classical Ramsey number. One of them is to determine Ramsey number R(G, H) for
general graphs G and H (not necessarily complete).
One of the most general results on graph Ramsey numbers is the establish-
ment of a general lower bound by Chv ́atal and Harary [17] which is formulated as:
R(G, H) ≥ (χ(H) − 1)(c(G) − 1) + 1, where G is a graph having no isolated vertices,
χ(H) is the chromatic number of H and c(G) denotes the cardinality of large con-
nected component of G.
Recently, Surahmat and Tomescu [41] studied the Ramsey number of a combina-
tion of path P n versus Jahangir graph J 2,m . They proved that R(P n , J 2,m ) = n+m−1
for m ≥ 3 and n ≥ (4m − 1)(m − 1) + 1. Furthermore, they determined that
R(P 4 , J 2,2 ) = 6 and R(P n , J 2,2 ) = n + 1 for n ≥ 5.
This dissertation studies the determination of Ramsey number for a combination
of path P n 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 P n with respect to Jahangir graph J 2,m . We improve the result of
Surahmat and Tomescu for m = 3, 4, 5 with n ≥ 2m + 1. Also, we determine the
Ramsey number for disjoint union of k identical copies of path P n versus Jahangir
graph J 2,m for m ≥ 2.
Moreover, we determine the Ramsey number of path P n versus generalized Ja-
hangir graph J s,m for different values of s, m and n. We also, evaluate the Ramsey
number for combination of disjoint union of t identical copies of path versus general-
ized Jahangir graph J s,m for even s ≥ 2 and m ≥ 3. At the end, we find the Ramsey
number of path versus beaded wheel BW 2,m , i.e. R(P n , BW 2,m ) = 2n − 1 or 2n if
m ≥ 3 is even or odd, respectively, provided n ≥ 2m 2 − 5m + 4.