The antimagic threshold characterization for unions with paths on three vertices

Let $G$ be a graph, let $P_3$ be the path on three vertices, and let $\tau(G)$ be the maximum integer such that the disjoint union of $G$ and $c$ copies of $P_3$ is antimagic for every integer $c$ with $0\le c\le\tau(G)$; if $G$ is not antimagic, set $\tau(G)=-\infty$. A graph is antimagic if it has a bijective edge labeling by $\{1,2,\ldots,|E(G)|\}$ whose vertex sums are pairwise distinct. Threshold characterization. For a graph $G$, the union of $G$ and $c$ copies of $P_3$ is antimagic if and only if $c\le\tau(G)$. This conjecture concerns the precise threshold for adding copies of $P_3$ to a graph; the supplied text gives no resolution status or further evidence beyond presenting it as a proposed statement.