2 problems

Let $G$ be a connected graph with at least three vertices. An edge weighting assigns a weight from $\{1,2,3\}$ to every edge of $G$, and the sum at a vertex is the sum of the weigh…

Let $G$ be a graph with maximum degree $\Delta$, and let $\chi”_{\sum}(G)$ denote the least number of colours in a proper total colouring whose weighted degrees distinguish adjacen…