Wheel and Star-critical Ramsey Numbers for Quadrilateral

The star-critical Ramsey number r∗(H1, H2) is the smallest integer k such that every red/blue coloring of the edges of Kn − K1,n−k−1 contains either a red copy of H1 or a blue copy of H2, where n is the graph Ramsey number R(H1, H2). We study the cases of r∗(C4, Cn) and R(C4, Wn). In particular, we prove that r∗(C4, Cn) = 5 for all n \u3e 4, obtain a general characterization of Ramsey-critical (C4, Wn)-graphs, and establish the exact values of R(C4, Wn) for 9 cases of n between 18 and 44

Star-critical Ramsey numbers and regular Ramsey numbers for stars

Let $G$ be a graph, $H$ be a subgraph of $G$, and let $G- H$ be the graph obtained from $G$ by removing a copy of $H$. Let $K_{1, n}$ be the star on $n+ 1$ vertices. Let $t\geq 2$ be an integer and $H_{1}, \dots, H_{t}$ and $H$ be graphs, and let $H\rightarrow (H_{1}, \dots, H_{t})$ denote that every $t$ coloring of $E(H)$ yields a monochromatic copy of $H_{i}$ in color $i$ for some $i\in [t]$. Ramsey number $r(H_{1}, \dots, H_{t})$ is the minimum integer $N$ such that $K_{N}\rightarrow (H_{1}, \dots, H_{t})$. Star-critical Ramsey number $r_{*}(H_{1}, \dots, H_{t})$ is the minimum integer $k$ such that $K_{N}- K_{1, N- 1- k}\rightarrow (H_{1}, \dots, H_{t})$ where $N= r(H_{1}, \dots, H_{t})$. Let $rr(H_{1}, \dots, H_{t})$ be the regular Ramsey number for $H_{1}, \dots, H_{t}$, which is the minimum integer $r$ such that if $G$ is an $r$-regular graph on $r(H_{1}, \dots, H_{t})$ vertices, then $G\rightarrow (H_{1}, \dots, H_{t})$. Let $m_{1}, \dots, m_{t}$ be integers larger than one, exactly $k$ of which are even. In this paper, we prove that if $k\geq 2$ is even, then $r_{*}(K_{1, m_{1}}, \dots, K_{1, m_{t}})= \sum_{i= 1}^{t} m_{i}- t+ 1- \frac{k}{2}$ which disproves a conjecture of Budden and DeJonge in 2022. Furthermore, we prove that if $k\geq 2$ is even, then $rr(K_{1, m_{1}}, \dots, K_{1, m_{t}})= \sum_{i= 1}^{t} m_{i}- t$. Otherwise, $rr(K_{1, m_{1}}, \dots, K_{1, m_{t}})= \sum_{i= 1}^{t} m_{i}- t+ 1$