On the Coalition Number of the d th Power of the n -Cycle

Qinglin Jia, Wenwei Zhao, Zhengyuan Jiang and Yongqiang Zhao ()
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Qinglin Jia: School of Mathematics and Physics, Hebei GEO University, Shijiazhuang 050031, China
Wenwei Zhao: Department of Electrical Engineering, University of South Florida, Tampa, FL 33620, USA
Zhengyuan Jiang: Computer Science and Engineering Department, University of South Florida, Tampa, FL 33620, USA
Yongqiang Zhao: School of Mathematics and Physics, Hebei GEO University, Shijiazhuang 050031, China

Mathematics, 2025, vol. 13, issue 11, 1-16

Abstract: A coalition in a graph G consists of two disjoint sets of vertices V 1 and V 2 , neither of which is a dominating set but whose union V 1 ∪ V 2 is a dominating set. A coalition partition in a graph G is a vertex partition π = { V 1 , V 2 , … , V k } such that every set V i ∈ π is not a dominating set but forms a coalition with another set V j ∈ π which is not a dominating set. The coalition number C ( G ) equals the maximum k of a coalition partition of G . In this paper, we study the coalition number of the d th power of the n -cycle C n d , where n ≥ 3 and d ≥ 2 . We show that C ( C n d ) = d 2 + 3 d + 2 for n = 2 d 2 + 4 d + 2 or n ≥ 2 d 2 + 5 d + 3 , and also provide some bounds of C ( C n d ) for the other cases. As a special case, we obtain the exact value of the coalition number of C n 2 .

Keywords: coalition; coalition partition; coalition number; dominating set (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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