A set S of vertices in a graph G is a dominating set of G if every vertex not in S has a neighbor in S, where two vertices are neighbors if they are adjacent. If G is isolate-free, then a set \(S \subseteq V(G)\) is a double dominating set of G if every vertex in \(V(G) \setminus S\) has at least two neighbors in S, and every vertex in S has at least one neighbor in S. A double coalition in G consists of two disjoint sets of vertices X and Y of G, neither of which is a double dominating set but whose union \(X \cup Y\) is a double dominating set of G. Such sets X and Y are said to form a double coalition. A double coalition partition in G is a vertex partition \(\Psi = \{V_1,V_2,\ldots ,V_k\}\) such that for all \(i \in [k]\) , the set \(V_i\) forms a double coalition with another set \(V_j\) for some j, where \(j \in [k] \setminus \{i\}\) . The double coalition number, \(\textrm{DC}(G)\) , of G equals the maximum order of a double coalition partition in G. We discuss the problem to determine or estimate the best possible constants \(\theta _{r}^{\textrm{reg}}\) and \(\theta _{r}\) (which depend only on r) for each \(r \ge 3\) , such that \(\textrm{DC}(G) \le \theta _{r}^{\textrm{reg}} \times r\) for the class of r-regular graphs G and \(\textrm{DC}(G) \le \theta _{r} \times \Delta (G)\) for the class of graphs G with minimum degree equal to r. We show that \(\theta _{r}^{\textrm{reg}} \ge 2 \left( \frac{r-1}{r} \right) \) for all \(r \ge 3\) , and that equality holds if \(r \in \{3,4\}\) , while \(\theta _{r} \ge 2\) for all \(r \ge 3\) , and \(\theta _{r} \ge 3\) for r sufficiently large. Moreover, we show that \(\theta _3 = 2\) . Finally, we prove that \(5 \le \textrm{DC}(G) \le 6\) whenever G is a 4-regular graph.
