On the Marking-tadpole and Marking-shell Ramsey numbers of graphs

Given two graphs G  and G2, the Ramsey number R(G ,G2) is the smallest integer N such that for any graph G of order N, either G1 is a subgraph of G, or G2 is a subgraph of the complement of G. Let Cn denote the cycle of order n and Pk be the path of order k. The Tadpole Graph Tn,k is the graph created by concatenating Cn and Pk with an edge from any vertex of Cn to a pendant of Pk for integers n = 3 and k = 0. A Shell Graph Sn,n−3 is a cycle graph with (n-3) chords sharing a common end point called the apex. De ined the Marking-Tadpole Graph Mp is a graph derived from Tadpole Graph Tn,k where every vertices are adjacent to every vertices of the graph. Moreover, the Marking-Shell Graph Jp is a graph originated from Shell Graph where every vertices are adjacent to every vertices of the graph as well. In this study, we show the combinatorial relationships and identities of R(Mp,C4) to R(Jp,C4) for every p ≥ 6 and the subscript p = n+3 of M.