Abstract
Let G be a graph with vertex set V and of order n=  V , and let δ(G) and Δ(G) be the minimum and maximum degree of G, respectively. 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 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 the coalition number C(G) of G. Given a coalition partition Ψ= { V_{1}, … , V_{k}} of G, a coalition graph CG (G, Ψ) is associated on Ψ such that there is a onetoone correspondence between its vertices and the members of Ψ, where two vertices of CG (G, Ψ) are adjacent if and only if the corresponding sets form a coalition in G. In this paper, we address previously some open problems in the study of the coalitions in graphs. We characterize all graphs G with δ(G) ≤ 1 and C(G) = n, and we characterize all trees T with C(T) = n 1. We determine the number of coalition graphs that can be defined by all coalition partitions of a given path. Furthermore, we show that there is no universal coalition path, a path whose coalition partitions define all possible coalition graphs.
Original language  English 

Article number  95 
Journal  Bulletin of the Malaysian Mathematical Sciences Society 
Volume  46 
Issue number  3 
DOIs  
Publication status  Published  May 2023 
Keywords
 Coalition graphs
 Coalition number
 Coalition partition
 Domination number
ASJC Scopus subject areas
 General Mathematics
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Data on Mathematical Science Reported by Researchers at Indian Institute for Technology (On the Coalition Number of Trees)
11/05/23
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