41 problems

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…

Let $\alpha_{n,m}$ denote the number of optimal solutions for the $n\times m$ grid in the paper's notation. In the case of a grid with $2$ rows and $2k+1$ columns, where $k\geq 0$,…

Let $SQ$ denote the infinite square grid, and let ERR:LD$(SQ)$ denote its error-correcting locating-dominating set density. Square-grid density conjecture. … The authors s…

Let $\mathbb{L}^{d}$ be the infinite $d$-dimensional square grid, whose vertices are the points of $\mathbb{R}^{d}$ with integer coordinates, with adjacency defined by Euclidean di…

Periodic perfect-power maximality conjecture. Its spanning-tree count is at most

Perfect-power square-maximality conjecture. Equality should occur only for the $d$-dimensional box $P_n^{\square d}$, up to lattice translation and coordinate permutation.

Let $f: \mathbb{N} \to \mathbb{N}$ be a strictly increasing function, let … for some $d>2$, and let $\mathcal G(f)=(G_n,n\geq 1)$. Write $P(\mathcal G(f))$ for the set of achievabl…

Let $m\geq n\geq 1$ be integers, and let $a$ denote the additional width parameter in a tree-decomposition of the $(n\times m)$-grid whose width is $n+a$. The spread versus width c…

Let $\text{TGr}_{m,n}$ be the toroidal grid graph formed from the Cartesian product of the cycle graphs $C_m$ and $C_n$, with $m,n\geq 3$, and let $\mathcal{R}(G)$ denote the repre…

Let $P_n$ be the path on $n$ vertices, let $P_n\mathbin{\square}P_n$ be the $n\times n$ Cartesian grid, and let $\operatorname{gp}_f$ denote the fractional general position number.…

Let $\mathcal{G}_{m\times n}$ denote the $m\times n$ grid graph, and let $\Delta_k(G)$ be the $k$-cut complex of a graph $G$. A simplicial complex is shellable if its facets admit…

Let $\mathcal{G}_{m\times n}$ denote the $m\times n$ grid graph, and let $\Delta_k^t(G)$ be the total $k$-cut complex of a graph $G$. For a simplicial complex $K$, write $b_i(K)$ f…

Let $P_m$ and $P_n$ be paths on $m$ and $n$ vertices, respectively, and let $P_m\square P_n$ be their Cartesian product, the $(m,n)$ grid graph. Let ${\rm P}{2}$ denote the second…

Let the $m \times n$ grid graph be partitioned into $k$ connected pieces, and consider the spanning tree distribution on partitions, in which a partition has weight proportional to…

Let $P_k$ be the path graph on $k$ vertices, let $d$ be a positive integer, and let $k_1,\ldots,k_d$ be positive integers. Write the Cartesian product grid as … Call a graph stacka…

Let $P_m$ and $P_n$ be paths on $m$ and $n$ vertices, respectively, and let $P_m\square P_n$ denote their Cartesian product. For a graph $G$, write $\sigma_{(p,q)}(G)$ for its…

Let $P_{2k+1}$ be the path graph on $2k+1$ vertices and $C_n$ the cycle graph on $n$ vertices. A graph is Non-Distance Magic (NDM) if it admits no bijective labeling…

Let $P_m$ be the path graph on $m$ vertices and $C_n$ the cycle graph on $n$ vertices. A graph is Non-Distance Magic (NDM) if it admits no bijective labeling…

Let $G_{2,m}$ be the rectangular grid with two rows and $m$ columns, and let $Ls(G_{2,m})$ and $Rs(G_{2,m})$ denote the corresponding left and right scores. Two-row grid score conj…

Let $G_{n,m}$ be the rectangular grid with $n$ rows and $m$ columns, with alternated black and white vertices and a black vertex in the top-left corner. Let $Ls(G_{n,m})$ and…

For positive integers $d$ and $t$, let $F_t^{(d)}$ be the $d$-dimensional grid with vertex set $[t]^d$, where two vertices are adjacent when they differ by exactly one in exactly o…

The conjecture. If

Interior-vertex lower-bound conjecture. For every interior vertex $X$,

Let $m\geq 2$, let ${\cal D}^{*}_m$ be the relevant digraph, and let ${\cal A}^{*}_m$ be its distinguished component. The maximum eigenvalue of a digraph means the largest eigenval…