Gallai-Ramsey Number for Rainbow S3

For the given graphs G and H , and for a positive integer k , the Gallai-Ramsey number is denoted by gr k ( G : H ) and is defined as the minimum integer n such that every coloring of the complete graph K n using at most k colors contains either a rainbow copy of G or a monochromatic copy of H . The k -color Ramsey number for G , denoted by R k ( G ) , is the minimum integer n such that every coloring of K n using at most k colors contains a monochromatic copy of G in some color. Let S n be the star graph on n edges and let P n be the path graph on n vertices. Denote by S + n the graph obtained from S n by adding an edge between any two pendant vertices. Let T n +2 be the tree on n + 2 vertices obtained from S n by subdividing one of its edges. In this paper, we consider gr k ( S 3 : H ) , where H ∈ { S n , S + n , P n , T n +2 } , and obtain its relation with R 2 ( H ) and R 3 ( H ) . We also obtain 3 -color Ramsey numbers for S n , S + n , and T n +2 .