Complete graphs and complete bipartite graphs without rainbow path

Motivated by Ramsey-type questions, we consider edge-colorings of complete graphs and complete bipartite graphs without rainbow path. Given two graphs G and H, the k-colored Gallai–Ramsey number gr _k (G:H) is defined to be the minimum integer n such that [Formula Presented] and for every N≥n, every rainbow G-free coloring (using all k colors) of the complete graph K _N contains a monochromatic copy of H. In this paper, we first provide some exact values and bounds of gr _k (P ₅ :K _t ). Moreover, we define the k-colored bipartite Gallai–Ramsey number bgr _k (G:H) as the minimum integer n such that n ² ≥k and for every N≥n, every rainbow G-free coloring (using all k colors) of the complete bipartite graph K _N,N contains a monochromatic copy of H. Furthermore, we describe the structures of complete bipartite graph K _n,n with no rainbow P ₄ and P ₅ , respectively. Finally, we find the exact values of bgr _k (P ₄ :K _s,t ) (1≤s≤t), bgr _k (P ₄ :F) (where F is a subgraph of K _s,t ), bgr _k (P ₅ :K _1,t ) and bgr _k (P ₅ :K _2,t ) by using the structural results.