A further result on the potential-Ramsey number of G1 and G2

Abstract

A non-increasing sequence ? = (d1,. . ., dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of ?. Given a graph H, a graphic sequence ? is potentially H-graphic if ? has a realization containing H as a subgraph. Busch et al. (Graphs Combin., 30(2014)847-859) considered a degree sequence analogue to classical graph Ramsey number as follows: for graphs G1 and G2, the potential-Ramsey number rpot(G1,G2) is the smallest non-negative integer k such that for any k-term graphic sequence ?, either ? is potentially G1-graphic or the complementary sequence ? = (k - 1 - dk,..., k - 1 - d1) is potentially G2-graphic. They also gave a lower bound on rpot(G;Kr+1) for a number of choices of G and determined the exact values for rpot(Kn;Kr+1), rpot(Cn;Kr+1) and rpot(Pn,Kr+1). In this paper, we will extend the complete graph Kr+1 to the complete split graph Sr,s = Kr ? Ks. Clearly, Sr,1 = Kr+1. We first give a lower bound on rpot(G, Sr,s) for a number of choices of G, and then determine the exact values for rpot(Cn, Sr,s) and rpot(Pn, Sr,s).