Ali, Kashif (2008) *On Ramsey numbers of path versus wheel-like graphs.* PhD thesis, Govt. College University, Lahore.

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## Abstract

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.

Item Type: | Thesis (PhD) |
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Subjects: | Physical Sciences (f) > Mathematics(f5) |

ID Code: | 2683 |

Deposited By: | Mr. Javed Memon |

Deposited On: | 03 Aug 2009 09:40 |

Last Modified: | 03 Aug 2009 09:40 |

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