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.
New bounds for locally irregular chromatic index of bipartite and subcubic graphs
一个图是\textit{locally irregular}，如果每个顶点$v$的邻居的度不同于$v$的度。图$G$的\textit{locally irregular edge-coloring}是一个（不适当的）边染色，使得在任何颜色类的边上导出的图是局部不规则的.证明了$3$颜色满足局部不规则边着色的要求。最近，Bensmail等人（Bensmail，Merker，Bensassen：Decomposing graphs into a constant number of local irregular subgraphs，{ \em European J. Combin.}，60：124--134，2017）将问题的第一恒定上限解决为$328$颜色。在本文中，使用现有的结果相结合，我们提出了一个改进的边界的二部图和一般图，设置最佳上界$7$和$220$，分别。此外，我们还证明了$4$色满足任何次立方图的局部不规则边染色。
