2 problems

Let $G$ be a graph and let $t$ be a positive integer. Write $G\cup tP_3$ for the disjoint union of $G$ and $t$ copies of $P_3$, and let $\tau(G)$ be the maximum integer such that…

Let $G$ and $H$ be antimagic graphs, and let $G\cup H$ denote their disjoint union. For an antimagic graph $X$, define $\tau(X)$ as the maximum integer such that $X\cup tP_3$ is an…