A study of two Ramsey numbers involving odd cycles

The \emph{book graph} of order $(n+2)$, denoted by $B_{n}$, is the graph with $n$ distinct copies of triangles sharing a common edge called the `base'. A cycle of order $m$ is denoted by $C_{m}$. A lot of studies have been done in recent years on the Ramsey number $R(B_{n}, C_{m})$. However, the exact value remains unknown for several $n$ and $m$. In 2021, Lin and Peng obtained the value of $R(B_{n}, C_{m})$ under certain conditions on $n$ and $m$. In this paper, they remarked that the value is still unknown for the range $n\in [\frac{9m}{8}-125, 4m-14]$. In a recent paper, Hu et al. determined the value of the book-cycle Ramsey number within the range $n\in [ \frac{3m-5}{2}-125, 4m]$ where $m$ is odd and $n$ is sufficiently large. In this article, we extend the investigation to smaller values of $n$. We have obtained a bound of $R(B_{n}, C_{m})$ if $n\in [2m-3, 4m-14]$ and $m\geq 7$ is odd. This is a progress on the earlier result. A connected graph $G$ is said to be \emph{$H$-good} if the formula, \begin{equation*} R(G,H)= (|G|-1)(\chi(H)-1)+\sigma(H) \end{equation*} holds, where $\chi(H)$ is the chromatic number of $H$ and $\sigma(H)$ is the size of the smallest colour class for the $\chi(H)$-colouring. In this article, we have studied the \emph{Ramsey goodness} of the graph pair $(C_{m}, \mathbb{K}_{2,n})$, where $\mathbb{K}_{2,n}$ is the complete biparite graph. We have obtained an exact value of $R(\mathbb{K}_{2,n},C_{m})$ for all $n$ satisfying $n\geq 3493$ and $n\geq 2m+499$ where $m\geq 7$ is odd. This shows that $\mathbb{K}_{2,n}$ is $C_{m}$-good, which extends a previous result on the Ramsey goodness of $(C_{m}, \mathbb{K}_{2,n})$. Also, this improves the lower bound on $n$ from a previous result on the Ramsey number $R(B_{n}, C_{m})$