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 a 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 prove that every isolate-free graph has a double coalition partition, and we show that \(2 \le \textrm{DC}(G) \le n\) and we characterize the graphs G satisfying \(\textrm{DC}(G) = 2\) and the graphs G satisfying \(\textrm{DC}(G) = n\) . We show that \(\textrm{DC}(G) \ge \delta (G) + 1\) where \(\delta (G)\) denotes the minimum degree among the vertices of G. We show that if \(\delta (G) \in \{1,2\}\) , then \(\textrm{DC}(G) \le \Delta (G) + 1\) where \(\Delta (G)\) denotes the maximum degree among the vertices of G. However we show that there exist graphs G with \(\delta (G) \ge 6\) satisfying \(\textrm{DC}(G) > \Delta (G) + 1\) . We determine the double coalition number of special classes of graphs. In particular, we determine the double coalition number of every cubic graph G and show that \(\textrm{DC}(G) = 4\) always holds.
