基于最小度条件的弱奇泛圈性

Weakly Odd Pancyclicity Threshold for the Minimum Degree

摘要: 奇围长是一个图中所有奇圈的长度的最小值。2024年，Yuan与Peng证明了如果一个$n$ 顶点非二部图$G$ 的最小度大于$n/6$ ，那么$G$ 包含从11到$2\lfloor n/21000\rfloor +1$ 之间的所有长度的奇圈。在本文中，我们证明了如果一个$n$ 顶点非二部图$G$ 的奇围长为$l$ 且最小度大于$n/\left(2\left(2l+1\right)\right)$ ，那么$G$ 包含从$2l+1$ 到$2\lfloor n/\left(128{\left(2l+1\right)}^4\right)\rfloor +1$ 之间的所有长度的奇圈。同时，这里的最小度条件是最好可能的。

Abstract: The odd girth of a graph is the minimum length of its odd cycles. In 2024, Yuan and Peng proved that if an $n$ -vertex non-bipartite graph $G$ has minimum degree greater than $n/6$ , then $G$ contains odd cycles of all lengths from 11 to $2\lfloor n/21000\rfloor +1$ . In this paper, we show that if an $n$ -vertex non-bipartite graph $G$ has odd girth $l$ and minimum degree greater than $n/\left(2\left(2l+1\right)\right)$ , then $G$ contains odd cycles of all lengths from $2l+1$ to $2\lfloor n/\left(128{\left(2l+1\right)}^4\right)\rfloor +1$ . Furthermore, the minimum degree condition is best possible.