112 problems

Let $G$ be a connected undirected graph. An antimagic orientation consists of an orientation $D$ of $G$ and a bijection from $E(G)$ to $\{1,2,\ldots,|E(G)|\}$ such that the vertex…

A tree is a connected graph with no cycles, and a labelling of a graph with $m$ edges is an injective map from its vertices to the positive integers. Such a labelling is graceful i…

An $L(2,1)$-labeling of a graph $G$ assigns nonnegative integers to its vertices so that labels of adjacent vertices differ by at least two and labels of vertices at distance two d…

Implicit Graph Conjecture. Every such class ${\mathcal F}$ admits an adjacency labeling scheme with labels of size $O(\log n)$.

Let $T$ be a tree, and let $P$ be a longest path in $T$. A tree is 2-distant, or a lobster, if every vertex of $T$ has distance at most $2$ from $P$. A tree is graceful if its vert…

Let $T$ be a tree with at least three vertices. An antimagic labeling is an edge labeling by the numbers $\{1,2,\dots,|E(T)|\}$ such that the sums of the labels of the edges incide…

Let $A$ be a finite Abelian group, and write $I(A)$ for the set of involutions of $A$. Set $A^*=A\setminus\{0\}$. An $A^*$-antimagic labeling of a graph with $|A|$ vertices is a bi…

Let $P_m$ be the path graph on $m$ vertices and $C_n$ the cycle graph on $n$ vertices, with $m,n\in\mathbb{N}$, $m\geq 2$, and $n\geq 3$. Their Cartesian product $P_m\Box C_n$ is t…

Let $G$ be a finite, simple, undirected graph, let $p$ be a positive integer, and let $\lambda^T_p(G)$ denote the minimum $k$ for which $G$ has a $(p,1)$-total $k$-labelling. Havet…

Let $G$ be a graph of size $m$, and let $es_g(G)$ denote its group edge irregularity strength, the least order of an Abelian group for which $G$ admits an edge-irregular labeling b…

Let $P_n$ denote the path graph on $n$ vertices, and let $P_n\times P_2$ be the Cartesian product, called the $n$-ladder graph. A prime labeling assigns the distinct integers in…

An antimagic labeling of a finite, undirected, simple graph with $m$ edges and $n$ vertices is a bijection from its edges to the integers $1,\ldots,m$ such that the sums of the lab…

Let a graph family be hereditary if it contains every induced subgraph of each of its graphs. Its speed is the function $s(n)$ giving the number of graphs on $n$ vertices in the fa…

A prime labeling of a graph $G$ is a bijection from $V(G)$ to $\{1,2,\ldots,|V(G)|\}$ such that adjacent vertices receive relatively prime labels. The tree prime-labeling conjectur…

A prism graph is the Cartesian product $P_2\square C_n$ of a path on two vertices and a cycle of length $n$. A graph is prime if its vertices can be bijectively labeled with…

Exact labeling-number conjecture.

Exact labeling-number conjecture.

Let $G$ be a finite, simple, undirected graph. A graph is $C$-antimagic if it admits an edge labelling whose vertex sums have at most $C$ conflicts, as defined in the paper. $C$-an…

For $X\subseteq\mathbb{N}$, let $K_X$ be the complete graph on vertex set $X$ in which each edge $ij$ has colour $|i-j|$. Write $[n]=\{1,\ldots,n\}$. A copy of a tree in this edge-…

Let $K_n$ be the complete graph on $n$ vertices. For an integer $m$, let $\mathbb{J}(n,m)$ be the set of primes $p\leqslant m$ for which $K_n$ is a Legendre cordial graph modulo…

Let $\Gamma(n,S)$ be a circulant graph on $n$ vertices with connection set $S$. Here, “small” means that $S$ is a connection set whose size is small relative to $n$, although no pr…

Let $G$ be a nice graph, and consider proper labellings of its edges by positive integers. The label-sum conjecture is that the minimum label sum over all proper labellings of $G$…

Let $G$ be a finite, simple, undirected graph of order $n$, maximum degree at most $\Delta$, and with no component of order $2$. A proper pushing scheme is a function…

Let $G$ be a graph. A total $k$-labelling is a map $\ell:V(G)\cup E(G)\to\{1,\dots,k\}$. For each vertex $u$, let $\mu_\ell(u)$ be the multiset of labels incident to $u$, including…

Let $G$ be a graph. A total $k$-labelling is a map $\ell:V(G)\cup E(G)\to\{1,\dots,k\}$. For each vertex $u$, let $\pi_\ell(u)$ be the product of the labels incident to $u$, includ…