论文标题
连接的公平分离超图
Connected Fair Detachments of Hypergraphs
论文作者
论文摘要
令$ \ Mathcal g $为一个边缘的超图。 $ \ mathcal g $的{\ it $（α，n）$ - 分离}是通过将顶点$α$分为$ n $ dertices（例如$α_1，\ dots，α_n$）获得的超图，并共享临界室中的事件铰链和evges。如果顶点的程度和边缘的多样性在整个超图和每个颜色类中的临界值之间尽可能均匀地共享，则分离是{\ it Fair}。在本文中，我们通过找到必要和充分的条件来解决70年代的开放问题，在该条件下，$ k $ - 颜色的HyperGraph $ \ Mathcal G $具有公平的分离，每个颜色类都可以连接。以前，当$ \ Mathcal g $是任意图（即2均匀的超图）时，这甚至不为人所知。我们通过证明有关超图解分解的各种新结果并完成部分常规组合结构，从而表现出我们定理的有用性。
Let $\mathcal G$ be a hypergraph whose edges are colored. An {\it $(α,n)$-detachment} of $\mathcal G$ is a hypergraph obtained by splitting a vertex $α$ into $n$ vertices, say $α_1,\dots,α_n$, and sharing the incident hinges and edges among the subvertices. A detachment is {\it fair} if the degree of vertices and multiplicity of edges are shared as evenly as possible among the subvertices within the whole hypergraph as well as within each color class. In this paper we solve an open problem from 70s by finding necessary and sufficient conditions under which a $k$-edge-colored hypergraph $\mathcal G$ has a fair detachment in which each color class is connected. Previously, this was not even known for the case when $\mathcal G$ is an arbitrary graph (i.e. 2-uniform hypergraph). We exhibit the usefulness of our theorem by proving a variety of new results on hypergraph decompositions, and completing partial regular combinatorial structures.
