A study of two Ramsey numbers involving odd cycles
Sayan Gupta
TL;DR
This work advances the study of Ramsey numbers involving book graphs and odd cycles by deriving a new bound for $R(B_n,C_m)$ in the regime $m\ge7$ odd and $n\in[2m-3,4m-14]$, showing $2n+3\le R(B_n,C_m)\le 2n+\tfrac{1010}{3}$. It also establishes an exact value for $R(\mathbb{K}_{2,n},C_m)$ when $n\ge 3493$ and $n\ge 2m+499$, $m$ odd, namely $R(\mathbb{K}_{2,n},C_m)=2n+3$, demonstrating the $C_m$-goodness of $\mathbb{K}_{2,n}$. The results leverage a toolkit of extremal and structural graph theory, including pancyclicity, bipanconnectedness, dependent random choice, and neighborhood intersection lemmas, to convert degree conditions into forced cycles in complements. A key distinction between the two main results lies in the presence of the base edge in $B_n$, which impedes a direct application of a crucial lemma used in the $K_{2,n}$ case. Overall, the paper narrows unknown ranges for these Ramsey numbers and sharpens the understanding of Ramsey goodness in this graph family.
Abstract
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)(χ(H)-1)+σ(H) \end{equation*} holds, where $χ(H)$ is the chromatic number of $H$ and $σ(H)$ is the size of the smallest colour class for the $χ(H)$-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 $R(B_{n}, C_{m})$