11 problems

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 $r\geq 3$ 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…

The product version of the 1-2-3 conjecture. Every nice graph $G$ admits a $3$-labelling such that $S_x$ is an independent set for every $x\geq 1$. This is a reformulation of the m…

Let $G$ be a graph with no component isomorphic to $K_2$; such a graph is called nice. Let $\chi_{\rm M}(G)$ denote the least number of labels needed for an m-proper edge-labelling…

Let $G$ be a graph with no component isomorphic to $K_2$; such a graph is called nice. For an edge-labelling $\ell:E(G)\to\{1,\dots,k\}$, define the sum at a vertex $v$ by … The la…

Let $G$ be a graph. A $k$-edge-labelling assigns to each edge a label in $\\{1,\dots,k\\}$, and the product at a vertex is the product of the labels on its incident edges. The mult…

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…