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, $rr(K_{ 1, m_{ 1}}, \dots, K_{ 1, m_{ t}})= \sum_{ i= 1}^{ t} m_{ i}- t+ 1$.
Categories: math.CO