11 problems

Let $r\geq 3$. Define … Two vertices are twins when the sets of edges containing them are equal, and a hypergraph is twin-free when it contains no twins. Kalkowski–Karoński–Pfender…

Let $r\geq 3$ and let $\mathcal{H}$ be an $r$-uniform hypergraph, meaning that every edge has size $r$. A hypergraph vertex coloring is proper when every edge contains at least two…

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 nice graph and let $w:V(G)\cup E(G)\rightarrow\{1,2,\ldots,k\}$ be a total integer weighting. For each vertex $v$, define … where $N(v)$ is the open neighborhood of…

Let $G$ be a nice graph, meaning a simple undirected graph with no component isomorphic to $K_2$. For an integer edge weighting $w:E(G)\rightarrow\{1,2,\ldots,k\}$, define … The we…

Let $G=(V,E)$ be a connected graph with at least two edges. A $3$-weighting of $G$ is a mapping $\omega:E\to\{1,2,3\}$, and it is vertex-colouring if the weighted degree…

Bača–Jendrol'–Miller–Ryan's conjecture. If $G\not\cong K_5$, then

Czerwiński–Grytczuk–Żelazny's conjecture. For every graph $G$,

Skowronek-Kaziόw's conjecture. For every graph $G$,

Total 1-2 Conjecture. For every graph $G$,

The 1-2 Conjecture. Every graph has a total $2$-weighting whose resulting vertex sums form a proper colouring; equivalently,