The join-graph characterization of local antimagic chromatic number

Let $G$ be a graph, let $O_n$ be the empty graph of order $n$, and let $G\vee O_n$ denote their join. Write $\chi(G)$ for the chromatic number and $\chi_{la}(G)$ for the local antimagic chromatic number.

Join characterization. For $n\ge 1$,

$$\chi_{la}(G\vee O_n)\ge\chi_{la}(G)+1$$

if and only if

$$\chi(G)=\chi_{la}(G).$$

The claim concerns when joining an empty graph increases the local antimagic chromatic number by at least one. The preceding discussion reports counterexamples to an earlier general lower bound and gives sufficient conditions for this characterization, but does not establish it in full.