9 problems

A tournament $R$ is a rebel if the class of tournaments not containing $R$ has bounded domination number. A poset tournament is a tournament admitting an ordering whose backedge gr…

For a tournament $T$, let $\operatorname{dom}(T)$ be its domination number and let $\operatorname{\overrightarrow{\omega}}(T)$ be its clique number. The domination-to-clique cluste…

Let $n\geq 3$ be odd, and let $G$ be a graph in $\mathbb{G}_{n,\lfloor n/2\rfloor}$. Write $\rho(G)$ for its spectral radius and $T^{(n+3)/2}_{(n-3)/2,0}$ for the specified graph i…

Minimum spectral radius conjecture. For every graph $G\in\mathbb{G}_{n,\lfloor\frac{n}{2}\rfloor}$,

Let $\mathcal{Q}(n)$ be the $n$-Queens' graph, and let $\gamma(\mathcal{Q}(n))$ denote its domination number. Cockayne's conjecture. For every $n \in \mathbb{N}$, … This asks wheth…

Let $\mathcal{Q}(n)$ be the $n$-Queens' graph, and let $\gamma(\mathcal{Q}(n))$ denote its domination number. For $k \in \mathbb{N} \cup \{0\}$, consider the order $n=4k+1$. Weakle…

Let $G_{k,2}$ be the bipartite graph with vertex classes $\binom{[n]}{k}$ and $\binom{[n]}{2}$, where a $k$-element set $S$ is adjacent to a 2-element set $T$ exactly when…

Let $\mathcal{P} \subseteq \mathcal{G}$ be additive and hereditary. A graph $G$ is vertex-$\gamma_{\mathcal{P}}$-critical if its vertex domination number with respect to…

Let $R$ be an arbitrary ring with Jacobson radical $J$, and suppose that … where each $F_i$ is a finite field and $n_i\geq 2$ for all $i$. For a finite ring $A$, let $\tau(A)$ deno…