82 problems

Let $p\geq 3$ be prime. A Hartke $\mathbb{Z}_p$-magic graph is a graph with the Hartke magic-labeling property over $\mathbb{Z}_p$; its order is its number of vertices. Low–Roberts…

Graham–Sloane conjecture. Every tree is harmonious.

An odd-graceful labeling is the tree-labeling notion in which the edge labels are the required distinct odd values. Odd-graceful tree conjecture. Every tree admits an odd-graceful…

Let $\overrightarrow{G}$ be a digraph of order $n$ whose weakly connected components all have order at least $3$, and let $\Gamma$ be an Abelian group. A $\Gamma$-irregular labelin…

Let $G=(V,E)$ be a finite simple undirected graph without isolated vertices. A super vertex total local antimagic labeling is a bijection $f:V\cup E\to\{1,2,\dots,|V|+|E|\}$ satisf…

Let $C_n$ and $C_m$ be cycles with $n,m\geq 3$. A supermagic labeling of a graph with $q$ edges is a bijection from its edge set to $\{1,2,\dots,q\}$ such that the sums of the labe…

Let $G$ be a graph. A $D$-neighborhood of a vertex is the set of vertices at distance at most $D$ from it. Neighborhood characterization conjecture. A graph is $D$-distance antimag…

Pinned-spine conjecture. Every FBT $T$ on $n$ vertices admits a graceful labeling $f$ such that, for some longest root-to-leaf path $v_0v_1\cdots v_\ell$, the restriction of $f$ to…

The harmonious labelling conjecture. For every $n$-edge tree $T$, there exists a labelling $\phi\colon V(T)\to \mathbb{Z}_n$ such that the edge sums $\phi(x)+\phi(y)$ are all disti…

Let $A$ be an Abelian group. A family of graphs is weakly $A$-cordial if all but finitely many graphs in the family are $A$-cordial. Erickson–Klein conjecture. For every Abelian gr…

Let $P_3$ be the path on three vertices. For a graph $H$, a $(1,1)$-antimagic labeling is an antimagic labeling whose vertex sums are exactly $1,2,\ldots,|V(H)|$. Pell-equation exi…

Let $G$ be a graph, let $P_3$ be the path on three vertices, and let $\tau(G)$ be the maximum integer such that the disjoint union of $G$ and $c$ copies of $P_3$ is antimagic for e…

Let $G$ be a graph, let $D\subseteq\{0,1,2,\dots,\operatorname{diam}(G)\}$ be a non-empty distance set, and define the $D$-neighborhood of a vertex $u$ by … A bijection…

Let $G=(V,E)$ be an undirected simple graph. A bijection $h:V(G)\rightarrow\{1,2,\dots,|V(G)|\}$ is a distance antimagic labeling when the vertex weights … are distinct for all ver…

Let $P_n$ be the path on $n$ vertices, and let $\gamma_e(P_n)$ denote its extended irregular domination number. Path-domination conjecture. For $n\geq 4$, … The equality is known f…

Let $C_{4n}$ be the cycle on $4n$ vertices, and let an optimal extended irregular dominating labeling mean a labeling satisfying the paper's extended irregular domination condition…

Let $\operatorname{\mathcal{T}}_{4,n}$ be the set of interlacing triangular arrays of rank $4$ and height $n$, and let $\square_n$ be the $n\times n$ square grid graph. An edge lab…

Let $G=(V,E)$ be a connected graph of size $m$. A labeling of its edges by $\{1,2,\dots,m\}$ is product antimagic if the products of the labels incident with the vertices are pairw…

A graph $G$ is 3-cordial if its vertices are labeled by $0$, $1$, and $2$ with label-class sizes differing by at most one, and each edge receives the sum of its endpoint labels mod…

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…

A family of graphs is weakly $A$-cordial if all but finitely many of its elements are $A$-cordial, where $A$ is an abelian group. Weak cordiality conjecture. For any abelian group…

A strongly graceful labeling is the strong graceful-labeling notion for trees used in the source. Broersma–Hoede conjecture. Every tree containing a perfect matching is strongly gr…

Let $T$ be a tree with a perfect matching. An adding-edge-subtracting dual graph transformation replaces an edge $u_{k-1}v_{k-1}\in E(H_{k-1})$ by an edge $x_ky_k\notin E(H_{k-1})$…

Let $G$ be a graph. A graph is prime if it admits a prime labeling, and odd prime if its vertices can be injectively labeled by $1,3,\ldots,2|V(G)|-1$ so that adjacent vertices rec…