无限度正则图的因子分解
Factorizations of regular graphs of infinite degree
摘要：
设$\mathcal{H}=\{H_i: i<α\}$是某个序数为$α$的图的指数族.图的$\mathcal{H}$ -分解$G$是$G$的边不交子图的族$\mathcal{G}=\{G_i: i<α\}$，使得对于每个$i<α$和$\bigcup\{E(G_i):i<α\}=E(G)$，$G_i$同构于$H_i$。$G$的$\mathcal{H}$ -因子分解是$G$的一个$\mathcal{H}$ -分解，使得$\mathcal{H}$的每个元素都是$G$的一个生成子图.让PDMATHAMV是一个无限基数。1936年，Kynnig证明了每一个$κ$ -正则图都有一个分解为完美匹配的因子。1980年，Andersen和Kassen利用这个定理证明了每个$κ$ -正则连通图都有一棵$κ$ -正则生成树。我们推广了这两个结果，并建立了对每个非零$λ\leq κ$，$κ$ -正则图到$λ$ -正则子图的因子分解的存在性。进一步证明了对于$κ$ -正则连通图，$κ$ -正则连通图的每一个$κ$森林族$\mathcal{H}$都有一个$\mathcal{H}$ -因子分解.
英文摘要：
Let $\mathcal{H}=\{H_i: i<α\}$ be an indexed family of graphs for some ordinal number $α$. $\mathcal{H}$-decomposition of a graph $G$ is a family $\mathcal{G}=\{G_i: i<α\}$ of edge-disjoint subgraphs of $G$ such that $G_i$ is isomorphic to $H_i$ for every $i<α$ and $\bigcup\{E(G_i):i<α\}=E(G)$. $\mathcal{H}$-factorization of $G$ is a $\mathcal{H}$-decomposition of $G$ such that every element of $\mathcal{H}$ is a spanning subgraph of $G$. Let $κ$ be an infinite cardinal. Kőnig in 1936 proved that every $κ$-regular graph has a factorization into perfect matchings. Andersen and Thomassen using this theorem proved in 1980 that every $κ$-regular connected graph has a $κ$-regular spanning tree. We generalize both these results and establish the existence of a factorization of $κ$-regular graph into $λ$-regular subgraphs for every non-zero $λ\leq κ$. Furthermore, we show that every $κ$-regular connected graph has a $\mathcal{H}$-factorization for every family $\mathcal{H}$ of $κ$ forests with $κ$ components of order at most $κ$ and without isolated vertices.
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