Extremal graphs for disjoint union of vertex-critical graphs

For a graph $F$, let $\mathrm{E}\mathrm{X}(n,F)$ be the set of $F$-free graphs of order $n$ with the maximum number of edges. The graph $F$ is called vertex-critical, if the deletion of its some vertex induces a graph with smaller chromatic number. For example, an odd wheel (obtained by connecting a vertex to a cycle of even length) is a vertex-critical graph with chromatic number 3. For $h\ge 2$, let $F_1,F_2,\mathrm{.}\mathrm{.}\mathrm{.},F_h$ be vertex-critical graphs with the same chromatic number. Let ${\cup }_{1\le i\le h}{F}_i$ be the disjoint union of them. In this paper, we characterize the graphs in $\mathrm{E}\mathrm{X}(n,{\cup }_{1\le i\le h}{F}_i)$, when there is a proper order among the graphs $F_1,F_2,\mathrm{.}\mathrm{.}\mathrm{.},F_h$. This solves a conjecture (on extremal problem for disjoint union of odd wheels) proposed by Xiao and Zamora \cite{XZ}.