Closure and Spanning k-Trees

Abstract

In this paper, we propose a new closure concept for spanning k-trees. A k-tree is a tree with maximum degree at most k. We prove that: Let G be a connected graph and let u and v be nonadjacent vertices of G. Suppose that $${\sum_{w \in S}d_G(w) \geq |V(G)| -1}$$ for every independent set S in G of order k with $${u,v \in S}$$ . Then G has a spanning k-tree if and only if G + uv has a spanning k-tree. This result implies Win’s result (Abh Math Sem Univ Hamburg, 43:263–267, 1975) and Kano and Kishimoto’s result (Graph Comb, 2013) as corollaries.