A total dominating set in a graph G is a set of vertices of G such that every vertex is adjacent to a vertex of the set. The total domination number γt(G) is the minimum cardinality of a dominating set in G. Thomassé and Yeo (2007) conjectured that if G is a graph on n vertices with minimum degree at least 5, then [Formula presented]. In this paper, it is shown that the Thomassé–Yeo conjecture holds with strict inequality if the minimum degree at least 6. More precisely, it is proven that if G is a graph of order n with δ(G)≥6, then [Formula presented]. This improves the best known upper bounds to date on the total domination number of a graph with minimum degree at least 6.
Bibliografisk noteFunding Information:
Research supported in part by the University of Johannesburg, South Africa.Research of the second author supported by the Danish research council under grant number DFF-7014-00037B.
© 2021 Elsevier B.V.