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A graph $\mathcal{H}=(W,E_\mathcal{H})$ is said to have {\em bandwidth} atmost $b$ if there exists a labeling of $W$ as $w_1,w_2,\dots,w_n$ such that$|i-j|\leq b$ for every edge $w_iw_j\in E_\mathcal{H}$. We say that$\mathcal{H}$ is a {\em balanced $(\beta,\Delta)$-graph} if it is a bipartitegraph with bandwidth at most $\beta |W|$ and maximum degree at most $\Delta$,and it also has a proper 2-coloring $\chi :W\rightarrow[2]$ such that$||\chi^{-1}(1)|-|\chi^{-1}(2)||\leq\beta|\chi^{-1}(2)|$.  In this paper, we prove that for every $\gamma>0$ and every natural number$\Delta$, there exists a constant $\beta>0$ such that for every balanced$(\beta,\Delta)$-graph $\mathcal{H}$ on $n$ vertices we have $$R(\mathcal{H},\mathcal{H}, C_n) \leq (3+\gamma)n$$ for all sufficiently large odd $n$. Theupper bound is sharp for several classes of graphs. Let $\theta_{n,t}$ be thegraph consisting of $t$ internally disjoint paths of length $n$ all sharing thesame endpoints. As a corollary, for each fixed $t\geq 1$, $R(\theta_{n,t},\theta_{n, t}, C_{nt+\lambda})=(3t+o(1))n,$ where $\lambda=0$ if $nt$ is oddand $\lambda=1$ if $nt$ is even. In particular, we have $R(C_{2n},C_{2n},C_{2n+1})=(6+o(1))n$, which is a special case of a result of Figaj and{\L}uczak (2018).

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