Critical Star Map Ramsey Number

Graph theory is an important part of discrete mathematics,It is an important branch of modern applied mathematics,Graph theory is widely used in many fields,such as physics,chemistry,operational research,computer science,information theory,cybernetics,network communication,social science and military,national defense,economic management,industrial and agricultural production.For two graphs G and H,the Ramsey number r(G,H)is the smallest positive integer r,so that the red G or the blue H will always be found by arbitrary red and blue coloring of the edges of Kr.Clearly,there must exists a critical graph Kr-1,which implies that there exists a critical red/blue edge coloring of Kr-1 that contains neither a red copy of G nor a blue copy of H.The star-critical Ramsey number r*(G,H)is the smallest positive integer k such that any red/blue coloring of the edges of graph Kr-1,∪Sk contains either a red subgraph that is isomorphic to G or a blue subgraph that is isomorphic to H where r=r(G,H).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.In 2015,Qizhong Lin and Yusheng Li obtained the Ramsey number of F2 versus Fn.Based on the known Ramsey Numbers,this paper studies and gives their star-critical Ramsey number.The paper is divided into four parts:The first part is the introduction,which introduces the historical background of Ramsey theory and some basic concepts of graph theory.Then it briefly introduces the research status of the star-critical Ramsey number and introduces the basic symbols involved in this paper.A general lower bound of the star-critical Ramsey number is given.This lemma is an important tool for proving the theorem in the following part.In the second part,we determine the star-critical Ramsey number of Fn versus K4(Note:This is the conclusion published in Discrete Applied Mathematics,but the original proof is incorrect,and this paper corrects the error).In the third part,we determine the star-critical Ramsey number of Fn versus F2.In the fourth part,we summarize and put forward some related problems.Finally,we give a brief introduction to the problems to be studied in the future.