Star-critical Ramsey Numbers For Large Generalized Fans And Books

Let G and H be simple graphs.The Ramsey number r（G,H）is defined to be the smallest positive integer r such that any red/blue edge coloring of Kr contains either a red copy of G or a blue copy of H.Clearly,there must exists a critical graph K_r－1 which implies that there exists a critical red/blue edge coloring of K_r－1 that contains neither a red copy of G nor a blue copy of H.The star-critical Ramsey number is the smallest integer k such that the induced graph contains a red G or a blue H by adding a new vertex that is adjacent to k vertices of K_r－1.The star-critical Ramsey number was first introduced by Hook and Isaak in 2010.They obtained some star-critical Ramsey numbers such as the star-critical Ramsey number for a tree of order n versus a complete graph.In1996,Li and Rousseau obtained the Ramsey number for large generalized fans versus complete graphs.Similarly,Nikiforov and Rousseau showed that large generalized books is Km-good in2004.In this thesis,the large part is devoted to the star-critical Ramsey number for large generalized fans and books.We shall show the Ramsey number for K₃ versus generalized F₃,n and the corresponding star-critical Ramsey number.In Chapter 1,we introduce some basic terminology of graph theory and Ramsey theory,then give the definition of the star-critical Ramsey number.Our main results are also listed in later part of this chapter.In Chapter 2,we introduce the regularity lemma and the stability lemma,and give a trivial lower bound for star-critical Ramsey number,which are powerful tools for subsequent proofs.In Chapter 3,we determine the star-critical Ramsey number for large gen-eralized fans and books.In Chapter 4,we determine the Ramsey number and star-critical Ramsey number of F₃,n versus K₃.In Chapter 5,we conclude what we have done and proposed some open problems that maybe similar to our work.