132 problems

Let $J_n(x)$ be a linear Jaco graph of order $n\geq1$. A diam-path is a path of diameter length, and let $P_d$ be a primary minimal dom-path, meaning a minimal path from $v_1$ to…

Let $P_k$ and $P_m$ be paths on $k$ and $m$ vertices, respectively, and let $G=P_k\mathbin{\square}P_m$ denote their Cartesian product. For a graph $G$, write ${\mathbb D}(G)$ for…

Let $J_\infty(x)$ be the infinite linear Jaco graph, and let $X$ be a $\gamma$-set, that is, a minimum dominating set. The vertex subscripts in the displayed set are…

Equality conjecture.

Let $G_{m,n}$ denote the $m$-by-$n$ grid graph, and let $\gamma_{\{2\}}(G_{m,n})$ be its integer $\{2\}$-domination number. The four-row grid conjecture. For every integer…

Let $G_{m,n}$ denote the $m$-by-$n$ grid graph, and let $\gamma_{\{2\}}(G_{m,n})$ be its integer $\{2\}$-domination number. The three-row grid conjecture. For every positive intege…

Tree upper-bound conjecture. If $3 \leq p \leq d$, then

Stability characterization conjecture. For every integer $p\ge 1$,

Dunbar et al.'s conjecture. If $G$ is a planar graph, then

Let $G$ be a finite simple undirected graph with vertex set $V(G)$, let $n=|V(G)|$, and let $d_k(G)$ denote the number of dominating sets of $G$ of size $k$. The polynomial … recor…

Let $G$ be a graph in $\mathcal{C}$ and let $v\in V(G)$ be such that $G-v\in\mathcal{C}$. Vertex-deletion conjecture. … The conjecture proposes the missing lower bound from the cit…

Let $T$ be a tree and let $k>2$. Write $\gamma_k(T)$ for the distance-$k$ domination number, $\gamma_{all,k}^{\infty}(T)$ for the eternal distance-$k$ domination number, and…

Mynhardt–Roux's conjecture. For every $n\geq3$, $P_n$ is not an $\operatorname{IR}$-graph, and for every $n\geq5$, $C_n$ is not an $\operatorname{IR}$-graph.

Let $G$ be a connected cubic graph, and let $V(G)$ denote its vertex set. A dominating set is a subset of vertices such that every vertex outside the subset has a neighbor in it. R…

Garijo, González and Márquez's conjecture. There exists an integer $n_1$ such that for any $n\geq n_1$, the maximum value of the location-domination number of a connected twin-free…

Let $M^n$ be an oriented closed manifold, where $n\ge 2$. A manifold is hyperbolic if it admits a Riemannian metric of constant negative curvature. Kotschick–Löh's conjecture. Ever…

Let $G$ and $H$ be graphs. Their strong product $G\boxtimes H$ has vertex set $V(G)\times V(H)$, with two vertices $(g_1,h_1)$ and $(g_2,h_2)$ adjacent when either $g_1=g_2$ and…

Let $C_n$ denote the directed cycle of order $n$, and let $b3_I(D)$ be the Italian domination number of a digraph $D$. For an odd integer $n$, Kim's conjecture. … This conjecture c…

Let $G$ be a graph. A broadcast $f$ has positive-support vertices $V_f^+=\{v\in V(G):f(v)>0\}$ and strength-one vertices $V_f^1=\{v\in V(G):f(v)=1\}$. Write…

Akbari et al.'s conjecture.

Let $G$ be a finite simple graph. It is factor-critical if, for every vertex $v\in V(G)$, the graph $G-v$ has a perfect matching. The graph $G$ is $K_{1,5}$-free if it has no induc…

Let $G$ be a cubic graph, with domination number $\gamma(G)$ and isolation number $\iota(G)$. Domination–isolation ratio conjecture. For every cubic graph $G$, … The conjecture wou…

Cornet–Dravec–Torres conjecture. For every odd integer $n\ge 7$,

Characterization problem. Characterize those trees $T$ for which

EISPN path-product conjectures. If $k \ge 1$, then