Ramsey and Gallai-Ramsey numbers for the union of paths and stars

Given k graphs H1, H2, ... , Hk, the k-color Ramsey number R(H1, H2, ... , Hk) is defined as the minimum integer n such that every k-edge-coloring of Kn contains a monochromatic Hi, for some i is an element of [k]. If H1 = H2 = ... = Hk = H, then we write the number as Rk(H). Given two graphs G and H, and a positive integer k, define the Gallai-Ramsey number grk(G : H) as the minimum number of vertices n such that any exact k-edge coloring of Kn contains either a rainbow copy of G or a monochromatic copy of H. Much like Ramsey numbers, Gallai-Ramsey numbers have gained a reputation as being difficult to compute in general. In this paper, we determine the exact values or upper and lower bounds of Gallai-Ramsey number grk(G : H) and Ramsey numbers R2(H), R3(H) when H is the union of a path and a star, and G is a 3-star or a 4-path or P4+, where P4+ is the graph consisting of a P4 with one extra edge incident with an inner vertex. (c) 2022 Elsevier B.V. All rights reserved.