A New Upper Bound on Total Domination Number of Bipartite Graphs

Let $ G $ be a graph. A subset $S subseteq V(G) $ is called a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $S$. The total domination number, $gamma_{t}$($G$), is the minimum cardinality of a total dominating set of $G$. In this paper using a greedy algorithm we provide an upper bound for $gamma_{t}$($G$), whenever $G$ is a bipartite graph and $delta(G)$ $geq$ $k$. More precisely, we show that if $k$ > 1 is a natural number, then for every bipartite graph $G$ of order $n$ and $delta(G) ge k$, $ $$gamma_{t}$($G$) $leq$ $n(1- frac{k!}{prod_{i=0}^{k-1}(frac{k}{k-1}+i)}).$