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Let G be a non-complete, finite simple graph with vertex set \(V(G)\) . We define the gathering number of G, denoted by \(g(G)\) , to be the minimum value of \(|X|-\omega (G-X)\) for all vertex cut-sets \(X\subset V(G)\) , where \(\omega (G-X)\) is the number of components of \(\langle G-X\rangle \) . We discuss the motivation behind the definition of the gathering number, and we provide conditions on the degree sequence of G which imply \(g(G)\geq k\) for a fixed integer k, with \(-|V(G)|\leq k\leq |V(G)|-4\) . We also indicate specifically how these degree conditions are best possible.

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The Gathering Number of a Graph and Vertex Degrees

  • 会议论文

  • 发表日期：

    2025年

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  • John Bazaar,
  • Michael Yatauro

摘要

Let G be a non-complete, finite simple graph with vertex set \(V(G)\) . We define the gathering number of G, denoted by \(g(G)\) , to be the minimum value of \(|X|-\omega (G-X)\) for all vertex cut-sets \(X\subset V(G)\) , where \(\omega (G-X)\) is the number of components of \(\langle G-X\rangle \) . We discuss the motivation behind the definition of the gathering number, and we provide conditions on the degree sequence of G which imply \(g(G)\geq k\) for a fixed integer k, with \(-|V(G)|\leq k\leq |V(G)|-4\) . We also indicate specifically how these degree conditions are best possible.