142 problems

Hovey's cordial labelling conjecture. Every tree admits a labelling by $\mathbb{Z}_k$, for every $k$, such that each vertex label occurs either $s$ or $s+1$ times and each edge lab…

Let $T$ be a tree with vertex set $V(T)$ and edge set $E(T)$, and let $n=|V(T)|$. A graceful labeling of $T$ is a bijection … such that the induced edge labels are exactly…

Graham–Sloane conjecture. Every tree is harmonious.

Let $G=(V(G),E(G))$ be a connected simple graph. A local antimagic labeling of $G$ is a bijection $f:E(G)\rightarrow\{1,2,\ldots,|E(G)|\}$ such that, writing … any two adjacent ver…

A cycle $C_n$ is a graph with $n$ vertices and $n$ edges, and a graph is perfect edge-magic if it admits an edge-magic labeling whose induced edge sums attain every possible valenc…

A graph $G$ with $m$ edges is graceful if there is an injection $f:V(G)\rightarrow\{0,1,\ldots,m\}$ such that the edge labels $|f(x)-f(y)|$, for edges $xy$, are pairwise distinct.…

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 $T$ 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 $P_n=v_1v_2\cdots v_n$ be the path of order $n$, and let $\chi_{lat}(P_n)$ denote its local antimagic total chromatic number, namely the minimum number of distinct vertex weigh…

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…

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…

A directed difference distance magic oriented graph (DDMOG) is an oriented graph equipped with a labeling function satisfying the paper's directed difference distance magic conditi…

Let $G(n,p)$) be an Erdős–Rényi random graph with constant edge probability $p$, and let $\operatorname{FR}_{\min}(G)$ denote the minimum Fourier ratio over all labelings of $G$. R…

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 $T\cong C(h_1,h_2,h_3)$ be a caterpillar of order $n$, and let $A$ be an Abelian group such that $|A|=n$ and $\exp(A)=m\geq 5$. Set … Here, $T$ is $A$-rainbow when it has an…

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 $\overrightarrow{G}$ 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 $P_n$ 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 $T$ be a tree. A gracious labeling is the labeling notion introduced for bipartite graphs in the cited literature, also called a near alpha-labeling in the surrounding discussi…

Let $T(A,B)$ be a bipartite tree with $n$ edges. A near alpha-labeling is a graceful labeling such that, for every edge $xy$ with $x\in A$ and $y\in B$, one has $f(x)<f(y)$. El-Zan…