The Ramsey numbers for disjoint unions of graphs

Abstract

For given graphs G and H , the Ramsey number R ( G , H ) is the smallest natural number n such that for every graph F of order n : either F contains G or the complement of F contains H . In this paper we investigate the Ramsey number of a disjoint union of graphs R ( ⋃ i = 1 k G i , H ) . For any natural integer k, we contain a general upper bound, R ( kG , H ) ⩽ R ( G , H ) + ( k - 1 ) | V ( G ) | . We also show that if m = 2 n - 4 , 2 n - 8 or 2 n - 6 , then R ( kS n , W m ) = R ( S n , W m ) + ( k - 1 ) n . Furthermore, if | G i | > ( | G i | - | G i + 1 | ) ( χ ( H ) - 1 ) and R ( G i , H ) = ( χ ( H ) - 1 ) ( | G i | - 1 ) + 1 , for each i , then R ( ⋃ i = 1 k G i , H ) = R ( G k , H ) + ∑ i = 1 k - 1 | G i | .