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.
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- Minimum degree six
- Total domination in graphs