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.
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Matsubara, R., Tsugaki, M. & Yamashita, T. Closure and Spanning k-Trees. Graphs and Combinatorics 30, 957–962 (2014). https://doi.org/10.1007/s00373-013-1314-z
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DOI: https://doi.org/10.1007/s00373-013-1314-z