A Degree Sequence Variant of Graph Ramsey Numbers

A (finite) sequence of nonnegative integers is graphic if it is the degree sequence of some simple graph G. Given graphs G1 and G2, we consider the smallest integer n such that for every n-term graphic sequence π, there is some graph G with degree sequence π with G1 ⊆ G or with G2 ⊆ G. When the phrase “some graph” in the prior sentence is replaced with “all graphs” the smallest such integer n is the classical Ramsey number r(G1, G2) and thus we call this parameter for degree sequences the potential-Ramsey number and denote it rpot(G1, G2). In this paper, we give exact values for rpot(Kn,Kt), rpot(Cn,Kt), and rpot(Pn,Kt) and consider situations where rpot(G1, G2) = r(G1, G2).