Closure and Spanning Trees with Bounded Total Excess

Abstract

Let \(\alpha \ge 0\) and \(k \ge 2\) be integers. For a graph G, the total k-excess of G is defined as \(\text{ te }(G;k)=\sum _{v \in V(G)}\max \{d_G(v)-k,0\}\). In this paper, we propose a new closure concept for a spanning tree with bounded total k-excess. We prove that: Let G be a connected graph, and let u and v be two non-adjacent vertices of G. If G satisfies one of the following conditions, then G has a spanning tree T such that \(\text{ te }(T;k) \le \alpha\) if and only if \(G+uv\) has a spanning tree \(T'\) such that \(\text{ te }(T';k) \le \alpha\):

  1. (i)

    \(\max \{ \sum _{x \in X} d_G(x): X \text{ is } \text{ a } \text{ subset } \text{ of } S \text{ with } |X|=k \} \ge |G|-1 \text{ for } \text{ every } \text{ independent } \text{ set } S \text{ in } G \text{ of } \text{ order } k+1 \text{ such } \text{ that } \{ u,v \} \subseteq S\); or

  2. (ii)

    \(\max \{ \sum _{x \in X} d_G(x): X \text{ is } \text{ a } \text{ subset } \text{ of } S \text{ with } |X|=k \} \ge |G|-\alpha -1 \text{ for } \text{ every } \text{ independent } \text{ set } S \text{ in } G \text{ of } \text{ order } k+\alpha +1 \text{ such } \text{ that } S \cap \{ u,v \} \ne \emptyset .\)

We also show examples to show that these conditions are sharp.

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Notes

  1. 1.

    We can generalize \(G_1\) by replacing each vertex in \(V_1\) with a complete graph.

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Acknowledgements

The authors would like to thank Professor Hikoe Enomoto, Professor Haruhide Matsuda and the anonymous referees for their valuable comments and suggestions. This work was supported by JSPS KAKENHI Grant number JP20J15332 (to S.M.), JP20K14353 (to T.Y).

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Correspondence to Shun-ichi Maezawa.

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Maezawa, Si., Tsugaki, M. & Yashima, T. Closure and Spanning Trees with Bounded Total Excess. Graphs and Combinatorics (2021). https://doi.org/10.1007/s00373-021-02283-z

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Keywords

  • Spanning tree
  • k-tree
  • k-ended tree
  • Total excess
  • Degree sum
  • Closure

Mathematics Subject Classification

  • 05C05
  • 05C50