Abstract In this note, we prove the following: Let G be a k -connected graph ( k ≥ 2 ) with circumference c ( G ) and m a non-negative integer. Then • [(1)] Either c ( G ) ≥ ( 2 m + 2 ) k , or d G ( v , C ) ≤ m for any longest cycle C and any vertex v of G . • [(2)] Either c ( G ) ≥ ( 2 m + 3 ) k , or d G ( e , C ) ≤ m for any longest cycle C and any edge e of G . When m = 0 , C in (1) and (2) are well-known Hamiltonian cycle and dominating longest cycle, respectively. Moreover, we give graphs to show that the bounds on c ( G ) are all sharp, even for those graphs that are triangle-free with only possible exception m = 0 in (2). (1) is also best possible for those graphs that are bipartite.

中文翻译：

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图的周长及其支配最长循环的距离

摘要 在本笔记中，我们证明以下内容： 设 G 是周长为 c ( G ) 且 ma 为非负整数的 ak 连通图 ( k ≥ 2 )。然后 • [(1)] 要么 c ( G ) ≥ ( 2 m + 2 ) k ，要么 d G ( v , C ) ≤ m 对于任何最长循环 C 和 G 的任何顶点 v。• [(2)] c ( G ) ≥ ( 2 m + 3 ) k ，或 d G ( e , C ) ≤ m 对于任何最长循环 C 和 G 的任何边 e。当 m = 0 时，(1) 和 (2) 中的 C 分别是众所周知的哈密顿循环和主导最长循环。此外，我们给出了图来表明 c ( G ) 的边界都是尖锐的，即使对于那些没有三角形的图，只有可能的例外 m = 0 在 (2) 中。(1) 对于二部图也是最好的。