4 problems

Let $G$ be a $d$-regular graph on $n$ vertices, and let $s(G)$ be the least integer $k$ for which the edges of $G$ can be assigned weights from $\\{1,\dots,k\\}$ so that all result…

1-2 Conjecture. Adjacent vertices of $G$ can be distinguished by these resulting sums using only the colors $\{1,2\}$.

Let a graph be $\{1,2\}$-weight colourable if its edges can be assigned weights from $\{1,2\}$ so that adjacent vertices receive distinct sums of the weights on their incident edge…

Let $G$ be a finite, undirected, simple connected graph with no isolated edge. A vertex-coloring $3$-edge-weighting assigns to each edge of $G$ a weight from $\{1,2,3\}$ so that, f…