Maximal Trees with Bounded Maximum Degree in a Graph

Abstract.

 Sein Win [5] obtained some degree conditions for the existence of a spanning tree with bounded maximum degree in a graph. Y. Caro, I. Krasikov and Y. Roditty [2], and, I. Krasikov and Y. Roditty [4] extended his results giving lower bounds for the size of largest tree with bounded maximum degree. We further extend the results by considering all trees with bounded maximum degree which are maximal with respect to the inclusion of vertex set. We then characterize the graphs which meet the bounds. Also, we obtain lower bounds for all maximal trees with bounded maximum degree in the graphs which satisfy a certain structure condition, as a generalization of G. Fan's condition [3] for hamiltonicity.