4 problems

Let $G$ be a graph, let $V(G)$ and $E(G)$ denote its vertex and edge sets, respectively, and let $k$ be an integer. An $(\iota,q)$-critical graph is a graph whose isolation number…

Let $G$ and $H$ be graphs, let $G\,\Box\,H$ denote their Cartesian product, and let $\iota(G)$ denote the isolation number of $G$. Isolation-number product conjecture. For any two…

Let $G$ be a cubic graph, and let $n(G)$ denote its order. Ziemann–Zyliński's conjecture. The bound … should hold for every cubic graph $G$ with $n(G)>6$, and would be sharp if tru…

Let $k\geq 3$, let $C_k$ be the cycle of length $k$, and let $G$ be a connected graph of size $m$ with $G\not\cong C_k$. For a set $D\subseteq V(G)$, write $N[D]$ for its closed ne…