On Star critical Ramsey numbers related to large cycles versus complete graphs

Abstract

Let K n denote the complete graph on n vertices and G, H be finite graphs without loops or multiple edges. Consider a two-coloring of edges of K n . When a copy of G in the first color, red, or a copy of H in the second color, blue is in K n , we write K n → (G, H). The Ramsey number r(G, H) is defined as the smallest positive integer n such that K n → (G, H). Star critical Ramsey r * (G, H) is defined as the largest value of k such that K r(G,H)−1 ⊔ K 1,k → (G, H). In this paper, we find r * (C n , K m ) for m ≥ 7 and n ≥ (m − 3)(m − 1).

Star-critical Gallai-Ramsey numbers involving the disjoint union of triangles

“…In a sense, it measures the strength of the corresponding Ramsey number. In their brief existence, star-critical Ramsey numbers have attracted the interest A c c e p t e d m a n u s c r i p t of numerous combinatorialists (e.g., see [2,8,12,13,14,15,18] and [20]). In a recent article, Su and Liu [17] considered the analogous concept for Gallai-Ramsey numbers.…”