Star-critical Ramsey numbers

Abstract

The graph Ramsey number R ( G , H ) is the smallest integer r such that every 2-coloring of the edges of K r contains either a red copy of G or a blue copy of H . We find the largest star that can be removed from K r such that the underlying graph is still forced to have a red G or a blue H . Thus, we introduce the star-critical Ramsey number r ∗ ( G , H ) as the smallest integer k such that every 2-coloring of the edges of K r − K 1 , r − 1 − k contains either a red copy of G or a blue copy of H . We find the star-critical Ramsey number for trees versus complete graphs, multiple copies of K 2 and K 3 , and paths versus a 4-cycle. In addition to finding the star-critical Ramsey numbers, the critical graphs are classified for R ( T n , K m ) , R ( n K 2 , m K 2 ) and R ( P n , C 4 ) .