Cooperative coloring of some graph families

In a family ${G_1, G_2, \ldots, G_m}$ of graphs sharing the same vertex set $V$, a cooperative coloring involves selecting one independent set $I_i$ from $G_i$ for each $i\in \{1,2,\ldots,m\}$ such that $\bigcup_{i=1}^m I_i = V$. For a graph class $\mathcal{G}$, let $m_{\mathcal{G}}(d)$ denote the minimum $m$ required to ensure that any graph family ${G_1, G_2, \ldots, G_m}$ on the same vertex set, where $G_i\in\mathcal{G}$ and $\Delta(G_i)\leq d$ for each $i\in \{1,2,\ldots,m\}$, admits a cooperative coloring. For the graph classes $\mathcal{T}$ (trees) and $\mathcal{W}$ (wheels), we find that $m_\mathcal{T}(3)=4$ and $m_\mathcal{W}(4)=5$. Also, we prove that $m_{\mathcal{B}^*}(d)=O(\log_2 d)$ and $m_{\mathcal{L}}(d)=O\left(\frac{\log d}{\log\log d}\right)$, where $\mathcal{B}^*$ represents the class of graphs whose components are balanced complete bipartite graphs, and $\mathcal{L}$ represents the class of graphs whose components are generalized theta graphs.