3 problems

Let $G$ be a graph without isolated edges. An edge coloring $f:E\to\{1,2,\dots,k\}$ assigns a weight $\sigma(v)=\sum_{v\in e}f(e)$ to each vertex; adjacent vertices are distinguish…

Let $G$ be a graph. A total coloring is a function $g:V\cup E\to\{1,2,\dots,k\}$, with total vertex weight $\sigma^t(v)=g(v)+\sum_{v\in e}g(e)$. Adjacent vertices are distinguished…

1-2-3 Conjecture. Every simple graph without an isolated edge has a neighbor-sum-distinguishing edge coloring using only the colors $1$, $2$, and $3$.