Abstract In 1963, Corradi and Hajnal settled a conjecture of Erdős by showing that every graph on at least 3 r vertices with minimum degree at least 2 r contains a collection of r disjoint cycles, and in 2008, Finkel proved that every graph with at least 4 s vertices and minimum degree at least 3 s contains a collection of s disjoint chorded cycles. The same year, a generalization of this theorem was conjectured by Bialostocki, Finkel, and Gyarfas: every graph with at least 3 r + 4 s vertices and minimum degree at least 2 r + 3 s contains a collection of r + s disjoint cycles, s of them chorded. This conjecture was settled and further strengthened by Chiba et al. (2010). In this paper, we characterize all graphs on at least 3 r + 4 s vertices with minimum degree at least 2 r + 3 s − 1 that do not contain a collection of r + s disjoint cycles, s of them chorded. In addition, we provide a conjecture regarding the minimum degree threshold for the existence of r + s disjoint cycles, s of them chorded, and we prove an approximate version of this conjecture.

中文翻译：

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图中具有给定最小度数的不相交环和弦环

摘要 1963 年，Corradi 和 Hajnal 解决了 Erdős 猜想，证明每个图在至少 3 r 个顶点上的最小度数至少为 2 r 的图都包含 r 个不相交圈的集合，2008 年，Finkel 证明了每个图4 s 顶点和至少 3 s 的最小度数包含 s 个不相交的弦循环的集合。同年，Bialostocki、Finkel 和 Gyarfas 推测了该定理的推广：每个具有至少 3 r + 4 s 个顶点且最小度数至少为 2 r + 3 s 的图包含一组 r + s 不相交循环，他们中的和弦。Chiba等人解决并进一步加强了这一猜想。(2010)。在本文中，我们在至少 3 r + 4 s 个顶点上刻画了所有图，这些顶点的最小度至少为 2 r + 3 s − 1，这些顶点不包含 r + s 不相交循环的集合，他们中的和弦。此外，我们提供了一个关于存在 r + s 个不相交循环的最小度阈值的猜想，其中 s 个是和弦的，我们证明了这个猜想的近似版本。