Let G be a graph with vertex set V and order \(n=|V|\) . A coalition in G is a combination of two distinct sets, \(A\subseteq V\) and \(B\subseteq V\) , which are disjoint and are not dominating sets of G, but their union \(A\cup B\) is a dominating set of G. A coalition partition of G is a partition \({\mathcal {P}}=\{S_1,\ldots , S_k\}\) of its vertex set V, where each set \(S_i\in {\mathcal {P}}\) is either a dominating set of G with only one vertex, or it is not a dominating set but forms a coalition with some other set \(S_j \in {\mathcal {P}}\) . The coalition number C(G) is the maximum cardinality of a coalition partition of G. To represent a coalition partition \({\mathcal {P}}\) of G, a coalition graph \(\textrm{CG}(G, {\mathcal {P}})\) is created, where each vertex of the graph corresponds to a member of \({\mathcal {P}}\) and two vertices are adjacent if and only if their corresponding sets form a coalition in G. A coalition partition \({\mathcal {P}}\) of G is a singleton coalition partition if every set in \({\mathcal {P}}\) consists of a single vertex. If a graph G has a singleton coalition partition, then G is referred to as a singleton-partition graph. A graph H is called a singleton coalition graph of a graph G if there exists a singleton coalition partition \({\mathcal {P}}\) of G such that the coalition graph \(\textrm{CG}(G,{\mathcal {P}})\) is isomorphic to H. A singleton coalition graph chain with an initial graph \(G_1\) is defined as the sequence \(G_1\rightarrow G_2\rightarrow G_3\rightarrow \cdots \) where all graphs \(G_i\) are singleton-partition graphs, and \(\textrm{CG}(G_i, \varGamma _1)=G_{i+1}\) , where \(\varGamma _1\) represents a singleton coalition partition of \(G_i\) . In this paper, we address two open problems posed by Haynes et al. We characterize all graphs G of order n and minimum degree \(\delta (G)=2\) such that \( C(G )= n\) . Additionally, we investigate the singleton coalition graph chain starting with graphs G, where \(\delta (G)\le 2\) .
