27 problems

Let $G$ be a connected graph that is not a cycle. Write $O(G)=V(G)\cup E(G)$, let $\operatorname{dim_m}(G)$ denote the minimum cardinality of a mixed metric generator of $G$, let…

Ghalavand et al.'s conjecture.

Let $K_n\square K_m$ be the Cartesian product of complete graphs, with $n\ge 3$, $m\ge n+1$, and $3\le k\le 2n$. The weak $k$-metric dimension formula asserts that … This conjectur…

Metric-dimension conjecture. The metric dimension of the supertoken graph ${\cal F}_k(K_n)$ is

Ghalavand's conjecture. The local metric dimension of $G$ satisfies

Metric dimension conjecture.

Odd-cycle conjecture. For every odd integer $n\ge5$,

Let $G$ be a connected graph with minimum degree $\delta(G)\geq2$. A daisy graph is a graph consisting of at least two cycles sharing one vertex, and a petal is one of its cycles;…

Let $G$ be a graph distinct from the cycle $C_n$, with minimum degree $\delta(G)\geq2$. Here $\mathrm{dim}(G)$ denotes the vertex metric dimension and $c(G)$ the cyclomatic number.…

Let $G$ be a connected graph. The invariant $L(G)$ is the graph parameter measuring the contribution of leaves, and $c(G)$ is the cyclomatic number of $G$. The vertex metric dimens…

The conjecture. If

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…

Leafless-graph upper-bound conjecture. Every leafless graph satisfies

Tree equidistant dimension conjecture.

Let $G$ be a connected graph. Let $L(G)$ denote the number of leaves, and let $c(G)=\left\vert E(G)\right\vert-\left\vert V(G)\right\vert+1$ be its cyclomatic number. The cycle-ran…

Let $G$ be a graph, let $c(G)$ be its cyclomatic number, let $L_1(G)$ be its number of leaves, and let $\mathrm{mdim}(G)$ be its mixed metric dimension. A cactus graph is a graph i…

Let $F_n$ be the folded $n$-cube obtained from the hypercube $Q_n$ by identifying each vertex $u$ with its antipode; write $\beta(F_n)$ for its metric dimension. Zhang's conjecture…

Let $G\ne C_n$ be a graph, and let $c(G)$ denote its cyclomatic number. The quantity $L_1(G)$ is the parameter used in the mixed metric dimension bound, and $\mathrm{mdim}(G)$ deno…

Cyclomatic-number bound. For every such graph,

Multiset-dimension upper-bound conjecture. If $G$ has finite multiset dimension, then

Let $G$ be a maximal planar graph with $n$ vertices, and let $\beta$ denote its metric dimension. The asymptotic metric-dimension conjecture. For maximal planar graphs, … The conje…

Complete bipartite strong resolving graph conjecture. The graph equation $G_{SR}\cong K_{r,s}$ has no solution for any $r,s\ge 2$.

Let $G$ be a graph. Write ${\rm \dim}(G)$ for its metric dimension, $\gamma(G)$ for its domination number, and $\psi(G)$ for its doubly resolving number. Metric-locating-dominating…

Let $H$ be a graph of order $n\ge 2$, let $K_1+H$ denote the graph obtained by adding a universal vertex to $H$, and let $\mathcal{C}(K_1+H)$ be its adjacency threshold. For…

Let $G$ be a graph. Write $\dim(G)$ for its metric dimension, $Z(G)$ for its zero forcing number, and $r(G)$ for its cycle rank. Cycle Rank Conjecture. … The conjecture proposes th…