All Ramsey critical graphs for large cycles vs a complete graph of order six

Abstract

A new area of Graph theory emerged in the last few decades is the calculation of Star critical Ramsey numbers related to different classes of graphs. Formally, we will say that Kn → (G,H) if given any coloring of Kn there is a copy of G in the first color, red, or a copy of H in the second color, blue. The Ramsey number r(G,H) is defined as the smallest positive integer n such that Kn → (G,H). A closely related concept of Ramsey number is the Star critical Ramsey number r∗(G,H) defined as the largest value of k such that Kr(G,H)−1 ⊔ K1,k → (G,H). A two-coloring of Kr(G,H)−1 such that Kr(G,H)−1 ̸→ (G,H) is called a Ramsey critical coloring. A Ramsey critical r(G,H) graph is a graph induced by the first color of a Ramsey critical coloring. Lower bounds for Star critical Ramsey numbers are usually found with the aid on Ramsey critical graphs. The particular problem we handle in this paper, on Star critical Ramsey numbers, is based on a conjecture posed in 1973 by Bondy and Erdos relating to Ramsey numbers for large cycles versus complete graphs. Based on certain lemmas we present with proof, furthermore we show that there exist exactly sixty eight non-isomorphic Ramsey critical r(Cn,K6) graphs, when n ≥ 15