General sharp upper bounds on the total coalition number
János Barát, Zoltán L. Blázsik · Discussiones Mathematicae Graph Theory · 2023
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 T C(G).We give a general sharp