9 problems

Let $G$ be a connected graph with at least three vertices. An edge weighting assigns a weight from $\{1,2,3\}$ to every edge of $G$, and the sum at a vertex is the sum of the weigh…

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…

Let $\mathcal{H}=(V,E)$ be a hypergraph, and call it $3$-uniform when every edge has size $3$. A hypergraph vertex coloring is proper when every edge contains at least two vertices…

Let $G$ be a graph, and call it nice if it has no connected component isomorphic to $K_2$. For an edge labelling $\ell$, let $\sigma_\ell(v)$ be the sum of the labels on the edges…

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…

The 1-2-3-Conjecture. For every graph $G$ without isolated edges, there is a weighting $\omega:E(G)\to\{1,2,3\}$ such that the induced vertex weights properly color $V(G)$.

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

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…

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