19 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…

Let $P_m$ be the path graph on $m$ vertices, let $C_n$ be the cycle graph on $n$ vertices, and let $P_m\Box C_n$ denote their Cartesian product. Write $\gamma(P_m\Box C_n)$ for its…

Let $G_{n,p}$ be the binomial random graph with edge-probability $p$, and let $\gamma(G_{n,p})$ denote its domination number, the smallest size of a set of vertices such that every…

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…

A tournament $H$ is a rebel if some constant bounds the domination number of every $H$-free tournament. The reverse of $H$ is obtained by reversing every edge. Rebel-reversal conje…

Let $G$ be a tournament, and let $\operatorname{dom}(G)$ denote its domination number. The reverse of a tournament is obtained by reversing every edge. Reverse-subtournament domina…

For tournaments $H_1,H_2,H_3$, let $\Delta(H_1,H_2,H_3)$ denote their cyclic composition. Define $\mathcal{S}_1$ to be the one-vertex tournament and, for $t\ge 2$, define…

For a tournament, the domination number is the minimum size of a set whose vertices dominate all vertices of the tournament. Bounded-small-subtournament domination conjecture. For…

For tournaments $H_1,H_2,H_3$, let $\Delta(H_1,H_2,H_3)$ be the tournament formed from disjoint sets inducing copies of $H_1,H_2,H_3$, with all edges directed cyclically between th…

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 $G$ be a connected nonbipartite graph on $n\geq 5$ vertices with domination number $\frac{n+1}{3}<\gamma\leq\frac{n-1}{2}$. Let $\mathbb{S}=\mathscr{H}_{3,\alpha}$ be a graph o…

Hening–Yeo conjecture. If $G\neq G_{14}$, then

Let $G$ be a connected graph. For a vertex $v$ in $G$, let $ecc_G(v)=\max\{d_G(v,x)\mid x\in V(G)\}$, let $diam(G)$ denote the diameter of $G$, and define the boundary of $G$ by ……

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…

Simplex domination-number conjecture. The domination number at $r=(d+1)/d$ satisfies