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arXiv 1602.02461math.CO
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不相交圈上Dirac和Erdés的加强定理

Strengthening theorems of Dirac and Erdős on disjoint cycles

Henry A. Kierstead, Alexandr V. Kostochka, Andrew McConvey

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13 pages
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摘要：

设$k \ge 3$为整数，$H_{k}(G)$为图$G$中度至少为$2k$的顶点集，$L_{k}(G)$为图$G$中度至多为$2k-2$的顶点集. 1963年，Dirac和Erdans证明了当$|H_{k}(G)| - |L_{k}(G)| \ge k^{2} + 2k - 4$时，$G$包含$k$（顶点-）不交圈.本文的主要结果是：对于$k \ge 2$，每个图$G$，若$|V(G)| \ge 3k$最多包含$t$不交三角形，若$|H_{k}(G)| - |L_{k}(G)| \ge 2k + t$最多包含$k$不交圈。这产生如果$k \ge 2$和$|H_{k}(G)| - |L_{k}(G)| \ge 3k$，则$G$包含$k$不相交的循环。这推广了Corrádi-Hajnal定理，该定理指出每个具有$H_{k}(G) = V(G)$和$|H_{k}(G)| \ge 3k$的图$G$包含$k$不相交的圈。

英文摘要：

Let $k \ge 3$ be an integer, $H_{k}(G)$ be the set of vertices of degree at least $2k$ in a graph $G$, and $L_{k}(G)$ be the set of vertices of degree at most $2k-2$ in $G$. In 1963, Dirac and Erdős proved that $G$ contains $k$ (vertex-)disjoint cycles whenever $|H_{k}(G)| - |L_{k}(G)| \ge k^{2} + 2k - 4$. The main result of this paper is that for $k \ge 2$, every graph $G$ with $|V(G)| \ge 3k$ containing at most $t$ disjoint triangles and with $|H_{k}(G)| - |L_{k}(G)| \ge 2k + t$ contains $k$ disjoint cycles. This yields that if $k \ge 2$ and $|H_{k}(G)| - |L_{k}(G)| \ge 3k$, then $G$ contains $k$ disjoint cycles. This generalizes the Corrádi-Hajnal Theorem, which states that every graph $G$ with $H_{k}(G) = V(G)$ and $|H_{k}(G)| \ge 3k$ contains $k$ disjoint cycles.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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