Abstract
For simple graphs G and H, their size Ramsey number is the smallest possible size of F such that for any red-blue coloring of its edges, F contains either a red G or a blue H. Similarly, we can define the connected size Ramsey number by adding the prerequisite that F must be connected. In this paper, we explore the relationships between these size Ramsey numbers and give some results on their values for certain classes of graphs. We are mainly interested in the cases where G is either a 2K 2 or a 3K 2, and where H is either a cycle Cn or a union of paths nPm . Additionally, we improve an upper bound regarding the values of and for certain t and m.
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