论文标题
将不规则的着色嵌入连接的因素化中
Embedding Irregular Colorings into Connected Factorizations
论文作者
论文摘要
对于$ r：=（r_1，\ dots，r_k）$，一个$ r $ - f骨化的$λ$ -fold $ h $ h $ h $ h $ -h $ siborm $ n $ n $ n $ n $ n $ vertex $λk_n^h $是（$λk_n^h $ fo_1 $ f_1 $ f_1 $ f_1 $ i = $ i = $ i = kkk $ i = kkk $ i. $ f_i $是$ r_i $ - regular和跨度。假设$ n \ geq（H-1）（2M-1）$。给定部分$ r $ - 物质化的$λk_m^h $，也就是说，$λk_m^h $的颜色（即分区）$ p $ to $ f_1，\ dots，f_k $，因此对于$ i = 1，\ dots，k $，k $，$ f_i $ ats $ for_i，$ f_i $ at $ f_ in $ f_i $ at $ span_i并确保$ p $的足够条件可以扩展到$λk_n^h $的连接$ r $ factorization（即，连接每个因素的$ r $ $ $ factorization）。此外，我们证明了一个一般结果，暗示了以下内容。给定任何子hypergraph的部分$ s $ factorization $ p $ $λk_m^h $，其中$ s：=（s_1，\ dots，s_q）$和$ q $并不太大，我们发现$ p $可以嵌入连接的$ r $ r $ -factorpare $ -Factorrization $ -Factorrization $ -Factorization $ -Factorization $ -Factorization $ -Factorization $ -Factorization $ -Factorization $ -Factorization $ -Factorization。这些结果可以看作是各种古典组合结果的统一概括，例如Cruse的嵌入部分对称拉丁正方形的定理，Baranyai的baranyai定理，关于超图的分解，希尔顿定理，希尔顿定理，将路径分解扩展到哈密顿分解，hägggkvist和Hellirem theorem theorem theorem theorem and theorem and theore theerem and-fercrized and-forcers。 Rodger和Wantland的定理嵌入了连接的因素化。
For $r:=(r_1,\dots,r_k)$, an $r$-factorization of the complete $λ$-fold $h$-uniform $n$-vertex hypergraph $λK_n^h$ is a partition of (the edges of) $λK_n^h$ into $F_1,\dots, F_k$ such that for $i=1,\dots,k$, $F_i$ is $r_i$-regular and spanning. Suppose that $n \geq (h-1)(2m-1)$. Given a partial $r$-factorization of $λK_m^h$, that is, a coloring (i.e. partition) $P$ of the edges of $λK_m^h$ into $F_1,\dots, F_k$ such that for $i=1,\dots,k$, $F_i$ is spanning and the degree of each vertex in $F_i$ is at most $r_i$, we find necessary and sufficient conditions that ensure $P$ can be extended to a connected $r$-factorization of $λK_n^h$ (i.e. an $r$-factorization in which each factor is connected). Moreover, we prove a general result that implies the following. Given a partial $s$-factorization $P$ of any sub-hypergraph of $λK_m^h$, where $s:=(s_1,\dots,s_q)$ and $q$ is not too big, we find necessary and sufficient conditions under which $P$ can be embedded into a connected $r$-factorization of $λK_n^h$. These results can be seen as unified generalizations of various classical combinatorial results such as Cruse's theorem on embedding partial symmetric latin squares, Baranyai's theorem on factorization of hypergraphs, Hilton's theorem on extending path decompositions into Hamiltonian decompositions, Häggkvist and Hellgren's theorem on extending 1-factorizations, and Hilton, Johnson, Rodger, and Wantland's theorem on embedding connected factorizations.
