61 problems

Let $G(n;k_{1},\dots,k_{n})$ be the super edge-magic total unicyclic $(p,q)$-graph consisting of an odd cycle … and $k_i$ pendant vertices adjacent to each $a_i$, for…

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 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…

Let $G$ be a simple graph. For a positive integer $k$, an edge $k$-labelling is a map $\ell:E(G)\to\{1,\dots,k\}$. For each vertex $u$, define its incident sum by … The labelling i…

Let $A$ be a finite Abelian group, and let $T$ be a tree. An $A$-antimagic labeling is an edge labeling whose induced vertex sums are pairwise distinct. Tree antimagic conjecture.…

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…

Cycle-star edge irregularity strength conjecture. For $k\geq 8$ and $n-k\geq 1$,

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…

Let $G$ be a graph with no isolated edge. A neighbor-sum-distinguishing (nsd) $3$-edge-weighting is a mapping from $E(G)$ to $\{1,2,3\}$ such that the sums of the weights incident…

Small Implicit Graph Conjecture. Every hereditary small graph class admits an implicit representation.

Let $G=(V,E)$ be a finite connected graph with no isolated vertices, and let $f:E\to\{1,2,\ldots,|E|\}$ be a bijection. For each vertex $u\in V$, define its weight by … where…

Let $G$ be a graph of order $n$ with no connected components of order less than $3$. A $\Gamma$-irregular labeling is a labeling of the edges of $G$ by elements of an abelian group…

A graph $G$ is harmonious when it admits a harmonious labeling, namely an injective vertex labeling whose induced edge labels are also injective modulo $|E(G)|$; for trees, the def…

Let $T$ be a nontrivial tree. A numbering of $T$ is a bijection $f:V(T)\to\{1,2,\ldots,|V(T)|\}$, and its strength is … The strength of $T$ is … where the minimum is over all numbe…

Let $G$ be a graph with maximum degree $\Delta$, and let $\lambda^{2,1}(G)$ be the minimum span of an $L(2,1)$-labeling, in which vertices at distance one receive labels differing…

Let $T$ be a tree with a perfect matching. An edge-mismatched transfer-operation removes an edge $uv$ from $T$ and adds a non-edge $xy$ when the edges $uv$ and $xy$ have the same e…

Let $G$ be a graph and let $t$ be a positive integer. Write $G\cup tP_3$ for the disjoint union of $G$ and $t$ copies of $P_3$, and let $\tau(G)$ be the maximum integer such that…

Let $C_n$ be a cycle of length $n\geq 3$, and let $C_n^t$ be the graph obtained from the union of $t$ copies of $C_n$ with one vertex in common, called the central vertex. A graph…

Random-graph clone conjecture. For every $\epsilon>0$, with probability $1$,