The Ramsey number for a cycle of length five vs. a complete graph of order six

Abstract

It has been conjectured that r(Cm, Kn) = (m − 1)(n − 1) + 1 for all m ≥ n ≥ 4. This has been proved recently for n = 4 and n = 5. In this paper, we prove that r(C5, K6) = 21. This raises the possibility that r(Cm, K6) = 5m − 4 for all m ≥ 5. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 99–108, 2000

All cycle‐complete graph Ramsey numbers r(Cm, K6)

“…3: Conjecture 1.1 was confirmed for n ¼ 3 in early work on Ramsey theory ( [6], [11]), and it has been proved recently for n ¼ 4 [13] and n ¼ 5 [2]. In [8] it has been proved that rðC 5 ; K 6 Þ ¼ 21: In this article, we will prove Conjecture 1.1 for n ¼ 6 and for all n ! 7 with m !…”

Star-critical Ramsey numbers for cycles versus K_4

“…The following four lemmas is a direct consequence of [8,9,7], written by Jayawardene et al The other three red graphs, namely R 15,1 , R 15,2 , R 15,3 , are illustrated in Figure 3.…”

“…The following four lemmas is a direct consequence of [8,9,7], written by Jayawardene et al Lemma 2). If G is a graph of order N that contains no C m and the independent number is less than or equal to n − 1 then the minimal degree is greater than or equal to N − r(C m , K n−1 ).…”

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

“…For ease of reference, we borrow the notation used in [6,8,9]. Given a graph G, we say Y ⊆ V (G) is an independent set if no pair of vertices of Y is adjacent to each other in G. Equivalently, Y forms a clique in G c .…”