18 problems

Let $G$ be a graph. A total list assignment gives each vertex and edge a list of allowable real weights; $G$ is $(k,k')$-choosable if, for every such assignment in which each verte…

Let $G$ be a graph with no isolated edge. A proper edge-weighting assigns weights to the edges of $G$ such that the sums of the weights on edges incident with adjacent vertices are…

Let $G=(V,E)$ be a simple graph. A $k$-edge-weighting is a function $\omega:E\rightarrow\{1,\ldots,k\}$. For each vertex $v\in V$, define its weighted degree by … A weighting is ve…

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 graph. A nice graph is one that contains no isolated edge. A vertex coloring $k$-edge weighting is a proper total weighting whose vertex weights are all zero and…

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…

Let $G=(V,E)$ be a graph without isolated edges. An edge $3$-weighting is a map $\omega:E\to\{1,2,3\}$, with weighted degree … at each vertex $v$. The weighting is vertex-colouring…

1-2-3 Conjecture. For every graph $G$ with no component isomorphic to $K_2$,

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,

The 1-2-3 Conjecture. There is an edge $3$-weighting such that $S$ is a proper colouring of $V(G)$; equivalently, any two adjacent vertices have different sums of incident edge wei…