In a family G1,G2,…,Gm of graphs sharing the same vertex set V, a cooperative coloring involves selecting one independent set Ii from Gi for each i∈{1,2,…,m} such that ⋃i=1mIi=V. For a graph class G, let mG(d) denote the minimum m required to ensure that any graph family G1,G2,…,Gm on the same vertex set, where Gi∈G and Δ(Gi)≤d for each i∈{1,2,…,m}, admits a cooperative coloring. For the graph classes T (trees) and W (wheels), we find that mT(3)=4 and mW(4)=5. Also, we prove that mB⁎(d)=O(log2d) and mL(d)=O(logdloglogd), where B⁎ represents the class of graphs whose components are balanced complete bipartite graphs, and L represents the class of graphs whose components are generalized theta graphs.