Three-color Ramsey number of an odd cycle versus bipartite graphs with small bandwidth

Abstract

A graph H = (W, E H ) is said to have bandwidth at most b if there exists a labeling of W as w 1 , w 2 , . . . , w n such that |i−j| ≤ b for every edge w i w j ∈ E H . We say H is a balanced (β, ∆)graph if it is a bipartite graph with bandwidth at most β|W | and maximum degree at most ∆, and it also has a proper 2-coloring χIn this paper, we prove that for every γ > 0 and every natural number ∆, there exist a constant β > 0 such that for every balanced (β, ∆)-graph H on n vertices we havefor all sufficiently large odd n. The inequality is asymptotically tight if H is a connected balanced (β, ∆)-graph on n vertices. Let θ n,t be the graph consisting of t internally disjoint paths of length n all sharing the same endpoints. As a corollary, for each fixed t ≥ 1, R(θ n,t , θ n,t , C nt+λ ) = (3t + o(1))n, where λ = 0 if nt is odd and λ = 1 if nt is even. In particular, R(C 2n , C 2n , C 2n+1 ) = (6 + o(1))n.