Some results related to the tenacity and existence of kk-trees

The tenacity of a graph GG, T(G)T(G), is defined by T(G)=min{|S|+τ(G−S)ω(G−S)}, where the minimum is taken over all vertex cutsets SS of V(G)V(G), ω(G−S)ω(G−S) be the number of components of G−SG−S and τ(G−S)τ(G−S) be the number of vertices in the largest component of the graph induced by G−SG−S.A kk-tree of a connected graph GG is a spanning tree with maximum degree at most kk. In this paper we show that if T(G)≥τ(G−S)ω(G−S)+1k−2, for any subset SS of V(G)V(G), with k≥3k≥3, then GG has a kk-tree.