a b s t r a c tThe 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(nK 2 , mK 2 ) and R(P n , C 4 ).

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A coprime labeling of a simple graph of order n is a labeling in which adjacent vertices are given relatively prime labels, and a graph is prime if the labels used can be taken to be the first n positive integers. In this paper, we consider when ladder graphs are prime and when the corresponding labeling may be done in a cyclic manner around the vertices of the ladder. Furthermore, we discuss coprime labelings for complete bipartite graphs.