线性超图的边染色

A Note on Edge Coloring of Linear Hypergraphs

中文关键词:  线性超图  边染色  Erd\H{o}s-Faber-Lov\'asz猜想

英文关键词:linear hypergraph  edge coloring  Erd\H{o}s-Faber-Lov\'asz conjecture

中文摘要:

      超图$H$的一个$k$-边染色是用$k$种颜色的边染色, 使得相交的边染不同的颜色. Erd\H{o}s-Faber-Lov\'asz猜想认为任一$n$个顶点的无环线性超图都有一个$n$-边染色. 2021年, Kang, Kelly, K\"uhn, Methuku和Osthus对充分大的$n$确认了该猜想成立. 在本文中, 我们证明该猜想对弱冲突的超图是成立的. 这严格拓展了Bretto, Faisant 和Hennecart在2020年的两个相关结果.

英文摘要:

      A $k$-edge coloring of a hypergraph $H$ is a coloring of the edges of $H$ with $k$ colors such that any two intersecting edges receive distinct colors. The Erd\H{o}s-Faber-Lov\'asz conjecture states that every loopless linear hypergraph with $n$ vertices has an $n$-edge coloring. In 2021, Kang, Kelly, K\"uhn, Methuku and Osthus confirmed the conjecture for sufficiently large $n$. In this paper, the conjecture is verified for collision-weak hypergraphs. This result strictly extends two related ones of Bretto, Faisant and Hennecart in 2020.