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

Abstract

Given k graphs H 1 , H 2 , … , H k , the k -color Ramsey number R ( H 1 , H 2 , … , H k ) is defined as the minimum integer n such that every k -edge-coloring of K n contains a monochromatic H i , for some i ∈ [ k ] . If H 1 = H 2 = … = H k = H , then we write the number as R k ( H ) . Given two graphs G and H , and a positive integer k , define the Gallai–Ramsey number gr k ( G : H ) as the minimum number of vertices n such that any exact k -edge coloring of K n 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 gr k ( G : H ) and Ramsey numbers R 2 ( H ) , R 3 ( H ) when H is the union of a path and a star, and G is a 3-star or a 4-path or P 4 + , where P 4 + is the graph consisting of a P 4 with one extra edge incident with an inner vertex.