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The cubic graphs with finite cyclic vertex connectivity larger than girth
Discrete Mathematics ( IF 0.7 ) Pub Date : 2021-02-01 , DOI: 10.1016/j.disc.2020.112197
Jun Liang , Dingjun Lou , Zan-Bo Zhang

Abstract Cyclic (vertex and edge) connectivity is an important concept in graphs. While cyclic edge connectivity ( c λ ) has been studied for many years, the study at cyclic vertex connectivity ( c κ ) is still at the initial stage. And c κ seems to be more complicated than c λ . We have got a sufficient condition that ν ( G ) ≥ 2 g ( k − 1 ) for c κ ≠ ∞ . On the other hand, if ν ( G ) 2 g ( k − 1 ) , then we have c κ = ∞ , or c κ ≤ ( k − 2 ) g , or ( k − 2 ) g c κ ∞ . So characterizing all the k -regular graphs with ( k − 2 ) g c κ ∞ is helpful to design an efficient algorithm for c κ . Hence, we characterize all 38 cubic graphs with g c κ ∞ and prove that c κ = g + 1 .

中文翻译:

有限循环顶点连通性大于周长的三次图

摘要 循环(顶点和边)连通性是图中的一个重要概念。虽然循环边连通性 (c λ ) 已经研究了很多年,但对循环顶点连通性 (c κ ) 的研究仍处于初级阶段。而 c κ 似乎比 c λ 更复杂。对于 c κ ≠ ∞,我们得到了 ν ( G ) ≥ 2 g ( k − 1 ) 的充分条件。另一方面,如果 ν ( G ) 2 g ( k − 1 ) ,那么我们有 c κ = ∞ ,或者 c κ ≤ ( k − 2 ) g ,或者 ( k − 2 ) gc κ ∞ 。因此用 (k − 2 ) gc κ ∞ 表征所有 k -正则图有助于设计一个有效的 c κ 算法。因此,我们用 gc κ ∞ 来表征所有 38 个立方图并证明 c κ = g + 1 。
更新日期:2021-02-01
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