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arXiv 2103.09745math.CO
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$C_k$平衡爆破子图的transmitted $C_k$ -因子

Transversal $C_k$-factors in subgraphs of the balanced blow-up of $C_k$

Beka Ergemlidze, Theodore Molla

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20 pages
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摘要：

对于图$F$的爆破的子图$G$，我们设$δ^*(G)$是由对应于$F$的边的部分对所诱导的$G$的所有二部子图上的最小度.在[三角形]中-平衡放大三角形中的因子。离散数学，2000]，Johansson证明了如果$G$是$C_3$的爆破的生成子图，其部分大小为$n$和$δ^*(G) \ge \frac{2}{3}n + \sqrt{n}$，则$G$包含$n$顶点不相交的三角形，并提出了Häggkvist的以下猜想：如果$G$是$C_k$的爆破的生成子图，其部分大小为$n$和$δ^*(G) \ge (1 + 1/k)n/2 + 1$，则$G$包含$C_k$的$n$顶点不相交副本，使得每个$C_k$与每个$k$部分恰好相交一次。当$k=3$时，该猜想的度条件是紧的，当$k \ge 4$ .时，该猜想的度条件不能被加强超过1，一个类似的猜想也作出了菲舍尔在[变种的Hajnal-Szemerédi定理。Journal of Graph Theory，1999]，Magyar和Martin在[Corrádi-Hajnal定理的三分版本]中证明了大$n$的三角形情况。离散数学，2002]。本文证明了这个猜想的渐近性。我们还提出了一个猜想，通过允许最小度条件的非空的二部子图诱导的部分对变化，推广了这一结果。我们的第二个结果支持这个新的猜想证明三角形的情况。这个结果渐近地推广了Johannson的结果。

英文摘要：

For a subgraph $G$ of the blow-up of a graph $F$, we let $δ^*(G)$ be the smallest minimum degree over all of the bipartite subgraphs of $G$ induced by pairs of parts that correspond to edges of $F$. In [Triangle-factors in a balanced blown-up triangle. Discrete Mathematics, 2000], Johansson proved that if $G$ is a spanning subgraph of the blow-up of $C_3$ with parts of size $n$ and $δ^*(G) \ge \frac{2}{3}n + \sqrt{n}$, then $G$ contains $n$ vertex-disjoint triangles, and presented the following conjecture of Häggkvist: If $G$ is a spanning subgraph of the blow-up of $C_k$ with parts of size $n$ and $δ^*(G) \ge (1 + 1/k)n/2 + 1$, then $G$ contains $n$ vertex disjoint copies of $C_k$ such that each $C_k$ intersects each of the $k$ parts exactly once. The degree condition of this conjecture is tight when $k=3$ and cannot be strengthened by more than one when $k \ge 4$., A similar conjecture was also made by Fischer in [Variants of the Hajnal-Szemerédi Theorem. Journal of Graph Theory, 1999] and the triangle case was proved for large $n$ by Magyar and Martin in [Tripartite version of the Corrádi-Hajnal Theorem. Discrete Mathematics, 2002]. In this paper, we prove this Conjecture asymptotically. We also pose a conjecture which generalizes this result by allowing the minimum degree conditions on the nonempty bipartite subgraphs induced by pairs of parts to vary. Our second result supports this new conjecture by proving the triangle case. This result generalizes Johannson's result asymptotically.

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