Geodesic
General sharp upper bounds on the total coalition number
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1567-1584

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Let G(V,E) be a finite, simple, isolate-free graph. Two disjoint sets A,B⊂ V form a total coalition in G, if none of them is a total dominating set, but their union A∪ B is a total dominating set. A vertex partition Ψ={C_1,C_2,...,C_k} is a total coalition partition, if none of the partition classes is a total dominating set, meanwhile for every i∈{1,2,...,k} there exists a distinct j∈{1,2,...,k} such that C_i and C_j form a total coalition. The maximum cardinality of a total coalition partition of G is the total coalition number of G and denoted by TC(G). We give a general sharp upper bound on the total coalition number as a function of the maximum degree. We further investigate this optimal case and study the total coalition graph. We show that every graph can be realised as a total coalition graph.
Keywords: total domination, total coalition partition, total coalition number, total coalition graph 
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Barát, János; Blázsik, Zoltán. General sharp upper bounds on the total coalition number. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1567-1584. http://geodesic.mathdoc.fr/item/DMGT_2024_44_4_a17/