General sharp upper bounds on the total coalition number

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 = {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