32 problems

Let $G_{n,\ell}:=K_{\underbrace{n,n,\ldots,n}_{\ell}}$ be the complete multipartite graph with $\ell$ parts, each of size $n$, and let $Z_q(G)$ denote the $q$-analogue of the zero…

Equality conjecture.

Let $K_{n,m}$ be the complete bipartite graph with parts of sizes $n$ and $m$, and let $K_2$ be the complete graph on two vertices. Write $Z_q(G)$ for the $q$-analogue of the zero…

Let $G$ and $H$ be graphs with an identified vertex $v$. Write G\,\makebox[0ex][l]{+} \raisebox{-.75ex}{\makebox[2ex]{{v}}}\,H for their vertex sum at $v$. Suppose that…

TxGraffiti's conjecture. If $G \not\cong K_4$ and $\Delta(G)\leq 3$, then

Let $G$ be a graph, let $n$ be its order, and let $\mathcal{F}_G$ denote the family of minimal forts of $G$. Let $\operatorname{Z}(G)$ denote the zero forcing number of $G$. Zero-f…

Let $G$ be a connected graph. A PSD fast join is a graph of order $n\geq 2$ such that $G\cong K_n$ or … for $t\geq 2$ and positive integers $n_i,m_i$ with $n=\sum_{i=1}^t n_i+m_i$.…

Cameron's conjecture. For every graph $G$,

Brimkov's conjecture. For every integer $d\geq 2$,

Propagation-time interval conjecture. For every $d\geq 2$,

Let $G$ be a graph. The parameters $Z(G)$ and $Z_+(G)$ denote the standard and positive semidefinite zero forcing numbers of $G$, respectively. A graph is claw-free if it has no in…

Boyer et al.'s conjecture. For every graph $G$ on $n$ vertices,

Let $G$ and $G'$ be graphs, each containing an edge. The Cartesian-product zero forcing conjecture. … The conjecture would extend the sharp lower bound known when the maximum nulli…

Davila–Henning zero-forcing conjecture.

Let $G_n=K(n,2)$ be the Johnson graph whose vertices are the $2$-subsets of an $n$-element set, with two vertices adjacent when the corresponding subsets intersect in one element.…

Let $G$ be an $n$-vertex graph, and let $z(G;k)$ denote the number of zero forcing sets of $G$ having size $k$. Let $P_n$ be the path on $n$ vertices. Fixed-size zero forcing conje…

Let $G$ be an $n$-vertex graph, let $P_n$ denote the path on $n$ vertices, and let $B_p(G)$ be the random subset of vertices obtained by including each vertex independently with pr…

For a graph $G$ on $n$ vertices, let $S(G)$ be the set of real symmetric matrices whose off-diagonal zero pattern is prescribed by adjacency in $G$. Define $mr(G)$ as the minimum r…

Hypercube leaky forcing conjecture.

Let $G$ be a connected graph. Write $Z(G)$ for its zero forcing number, let $\dim(G)$ denote its metric dimension, and let $c(G)$ be the number of edges that must be removed from…

Let $G$ be a connected graph. Let $Z(G)$ denote its zero forcing number, let $c(G)=\left\vert E(G)\right\vert-\left\vert V(G)\right\vert+1$ be its cyclomatic number, and let…

Let $t$ be an integer satisfying $t\equiv 0,1,2\pmod{6}$, and let $r$ be an integer with $r>\lfloor t/6\rfloor$. The extended cube graphs are denoted by…

Let $k$ be a positive integer and let [... ELLIPSIZATION...]rname{Circ}[(\ell^2-1)k,\{1,\ell\}])=2\ell. \] and field independent minimum rank with universally optimal matrix…

For the $n\operatorname{-Sun}$ graph, let $\operatorname{ept}(n\operatorname{-Sun})$ denote its expected propagation time. Limiting increment conjecture. … The conjecture is motiva…

Let $K_{m,n}$ have partite vertex sets $R$ and $R'$ of orders $m$ and $n$ respectively, and let $u\in R$ and $v\in R'$. Partite-set comparison conjecture. If $n>3$ and $m<n$, then…