Star-critical Ramsey numbers for cycles versus K_4

Abstract

Let G, H and K represent three graphs without loops or parallel edges and n represent an integer. Given any red blue coloring of the edges of G, we say that K → (G, H), if there exists red copy of G in K or a blue copy of H in K. Let K n represent a complete graph on n vertices, C n a cycle on n vertices andWhen n > 3, in this paper we show that r * (C n , K 5 ) = 3n − 1 except r * (C 4 , K 5 ) = 13. We also characterize all Ramsey critical r(C n , K 5 ) graphs.

Multicolor star-critical Ramsey numbers and Ramsey-good graphs

“…Here, K r−1 K 1,k is the graph formed by taking the disjoint union of K r−1 with a vertex v, then adding edges between v and exactly k of the vertices in the K r−1 . Star-critical Ramsey numbers in the case where t = 2 have been extensively studied (e.g., see [24], [27], [28], [29], [33], [34], [36], [42], and [44]).…”

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.…”