6 problems

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, 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 $G$ 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 graph, and let $\chi(G)$ denote its chromatic number. For a vertex decoration $f$, define $S(v)=\sum_{x\in N(v)}f(x)$, and call $f$ cool if $S(u)\ne S(v)$ for every ad…

Let $G$ be a connected graph of order $n$. The path extremal conjecture. … Here $vcfc(G)$ denotes the conflict-free vertex-connection number of $G$, and $P_n$ is the path on $n$ ve…