16 problems

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…

1-2-3 Conjecture. Every finite graph without an isolated edge has a 3-weighting.

Let $G=(V,E)$ be a simple graph. A vertex-coloring edge-weighting is an edge-weighting $\omega:E\to\\{1,2,3\\}$ such that, writing … we have $s_{\omega}(u)\neq s_{\omega}(v)$ for e…

Let $G$ be a simple graph on $n$ vertices with minimum degree $\delta\geq 1$ and no isolated edges. Define the irregularity strength $s(G)$ as the least positive integer $k$ for wh…

A graph $G$ has no isolated edges or isolated triangles if no connected component is isomorphic to $K_2$ or $K_3$. A subgraph fulfills the 1–2–3 Conjecture when it admits a neighbo…

A graph $G=(V,E)$ has no isolated edges if none of its connected components is a single edge. A weighting $cspanomega:E\to\{1,2,3\}$ is sum-distinguishing if the weighted degrees ……

Let $G$ be a nice graph, meaning a graph with no connected component isomorphic to $K_2$. A $k$-edge-weighting assigns to each edge a weight from $\{1,\ldots,k\}$, and…

Let $G$ be a nice graph, meaning a graph with no connected component isomorphic to $K_2$. An edge-weighting is edge-injective if it assigns distinct weights to all edges, and…

Let $G$ be a graph with no isolated edges. A vertex colouring $3$-edge weighting is a proper edge weighting assigning to every edge a value in $\{1,2,3\}$. The 1-2-3 conjecture. Ev…

The 1-2-3 Conjecture for 3-uniform hypergraphs. There is such a weighting for which the induced vertex weights properly color $V(H)$.

Hypergraph 1-2-3 Conjecture. There is such a weighting for which the induced vertex weights properly color $V(H)$.

The 1-2-3 Conjecture. There is such a weighting for which the induced vertex weights properly color $V(G)$.

Let $G$ be a graph with no component isomorphic to $K_2$. An edge $k$-list-weighting assigns to each edge a weight from an independently assigned list of $k$ real numbers. Let…

Let $G$ be a graph with no component isomorphic to $K_2$. An edge $k$-weighting assigns to every edge a number from $[k]=\{1,2,\ldots,k\}$. The sum at a vertex is the sum of the we…

1-2-3-conjecture. For every connected graph $G$ with at least three vertices,

A finite, undirected, simple connected graph $G$ is called nice if it has no component isomorphic to $K_2$. A $k$-edge-weighting assigns to every edge $e$ an integer weight…