A new upper bound is given for the cycle-complete graph Ramsey number r(C,,,, K"), the smallest order for a graph which forces it to contain either a cycle of order m or a set of ri independent vertices . Then, another cycle-complete graph Ramsey number is studied, namely r( :C,,,, K") the smallest order for a graph which forces it to contain either a cycle of order 1 for some I satisfying 3 :1 !~ ,m or a set of n independent vertices . We obtain the exact value of r( :C,,,, K") for all m > n and an upper bound which applies when m is large in comparison with log n .
In a seminal paper from 1983, Burr and Erdős started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemerédi regularity lemma, embedding of sparse graphs, Turán type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving five -all but one -of the Burr-Erdős problems.A weaker version of this conjecture was stated earlier by Burr in [8].Conjecture 1.2. For fixed q ≥ 1, p ≥ 3, all sufficiently large graphs of maximum degree at most q are p-good. Brandt [4] showed that for p = 3 and q ≥ 168, every q-regular graph of sufficiently large order and with sufficiently large expansion factor is a counterexample to Conjecture 1.2. In [27] we showed that, for p = 3, almost all 100-regular graphs are counterexamples to Conjecture 1.2, and thus to Conjecture 1.1.We shall answer in the affirmative all but one of the remaining questions raised in [7].Write C n for the cycle of order n. Burr and Erdős [7] showed that the wheel K 1 +C n is 3-good for n ≥ 5. This result motivated the following three questions ([7], p. 50). Question 1.3. Is the wheel K 1 + C n p-good for fixed p > 3 and n large?Recall that the kth power of a graph G is a graph G k with V (G k )= V (G) and uv ∈ E(G k ) if u and v can be joined in G by a path of length at most k. Question 1.4. Is K 1 +C k n a p-good graph for fixed k ≥ 2, p ≥ 3 and n large?
For n = 1, 2, . . . , let 6, = K2+ K,,. We pose the problem of determining the Ramsey numbers r (&, B,) and demonstrate that in many cases critical colorings are available from known examples of strongly regular graphs.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.