Degree Ramsey numbers for even cycles

Let H -> G denote that any s-coloring of E(H) contains a monochromatic G. The degree Ramsey number of a graph G, denoted by R Delta(G, s), is min{Delta(H) : H -> G}. We consider degree Ramsey numbers where G is a fixed even cycle. Kinnersley, Milans, and West showed that R-Delta(C-2k, s) >= 2s, and Kang and Perarnau showed that R-Delta(C-4, s) = Theta(s(2)). Our main result is that R-Delta(C-6, s) = Theta(S-3/2) and R-Delta(C-10, s) = Theta(S-5/4). Additionally, we substantially improve the lower bound for R-Delta(C-2k, s) for general k. (C) 2017 Elsevier B.V. All rights reserved.