On the coalition number of graphs
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Let G be a graph with vertex set V. Two disjoint sets V_1, V_2 ⊆ V form a coalition in G if none of them is a dominating set of G but their union V_1∪ V_2 is. A vertex partition Ψ={V_1,…, V_k} of V is called a coalition partition of G if every set V_i∈Ψ is either a dominating set of G with the cardinality |V_i|=1, or is not a dominating set but for some V_j∈Ψ, V_i and V_j form a coalition. The maximum cardinality of a coalition partition of G is called the coalition number of G, denoted by 𝒞(G). A 𝒞(G)-partition is a coalition partition of G with cardinality 𝒞(G). Given a coalition partition Ψ={V_1, V_2,…, V_r} of G, a coalition graph CG(G, Ψ) is associated on Ψ such that there is a one-to-one correspondence between its vertices and the members of Ψ. Two vertices of CG(G, Ψ) are adjacent if and only if the corresponding sets form a coalition in G. In this paper, we first show that for any graph G with δ(G)=1, 𝒞(G)≤ 2Δ(G)+2, where δ(G) and Δ(G) are the minimum degree and the maximum degree of G, respectively. Moreover, we characterize all graphs G with δ(G)≤ 1 and 𝒞(G)=n, where n is the number of vertices of G. Furthermore, we characterize all trees T with 𝒞(T)=n and all trees T with 𝒞(T)=n-1. This solves partially one of the open problem posed in <cit.>. On the other hand, we theoretically and empirically determine the number of coalition graphs that can be defined by all coalition partitions of a given path P_k. Furthermore, we show that there is no universal coalition path, a path whose coalition partitions defines all possible coalition graphs. These solve two open problems posed by Haynes et al. <cit.>.
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