Gallai-Ramsey Numbers For Some Small Graphs

Ramsey theory started in the late 1920s.It was originally proposed by the British mathematician F.P.Ramsey.From then on,it was increasingly understood and devel-oped.The meaning of Ramsey theory is clear:Complete disorder is impossible.By combinatorial words,every large set of numbers,points or objects necessarily contains a given substructure.Ramsey number is one of the central problems in Ramsey the-ory.The definition of Ramsey numbers is as follows:for any positive integers k and l,the Ramsey number r（k,l）is the smallest positive integer n,such that every graph on n vertices contains either a clique of k vertices or a stable set of l vertices.In other words,the Ramsey number r（k,l）is the smallest positive integer n,such that for any 2-edge-coloring of Kn（by red and blue）,there is a red Kk or a blue Kl.According to the following definition,Ramsey numbers can be generalized to generalized Ramsey numbers.Given an integer k≥1 and graphs H₁,…,H_k,Ramsey number r（H₁,…,H_k）is defined to be the minimum integer n,such that for any k-coloring of the edges of Kn,there is a monochromatic copy of Hi in color i for some integer i（1≤i≤k）.In this work,we study Ramsey numbers of graphs in Gallai colorings,where a Gallai coloring is a coloring of the edges of a complete graph without rainbow trian-gles（that is,a triangle with all its edges colored differently）.Gallai-Ramsey num-ber grk（K₃:H₁,H₂,…,H_k）is defined as follows:given an integer k and graphs H₁,…,H_k,grk（K₃:H₁,H₂,…,H_k）is the smallest positive integer n,such that for any k-coloring of the edges of Kn,there is either a rainbow triangle,or a monochro-matic copy of Hi in color i for some integer i（1≤i≤k）.If H₁=H₂=…=Hs=H,H_s+1=H_s+2=…=H_k=G,the Gallai-Ramsey number is written as grk（K₃:sH,（k-s）G）.If H₁=H₂=…=H_k=H,the k-colored Gallai-Ramsey number for H is written as grk（K₃:H）.In this paper,we focus on the Gallai-Ramsey numbers for K4-e versus triangles and K₃+e versus triangles.For Gallai-Ramsey numbers of K4-e versus triangles,we prove the following result:Theorem 1 If k≥4 and s is an integer with 0≤s≤k,then grk（K₃:s（K4-e）,（k-s）K₃）=g（k,s）where#12 For Gallai-Ramsey numbers of K₃+e versus triangles,we prove the following results:Theorem 2 For s=1,2,3,gr3（K₃:s（K₃+e）,（3-s）K₃）=16.Theorem 3 Given integers k≥3,#12 For the Gallai-Ramsey numbers grk（K₃:s（K₃+e）,（k-s）K₃）and k>3,we only get the result when s=1.For k>3 and s≥2,we have the following conjecture:Conjecture 4 Given integers k≥3 and s≥2,（?）...