A linearly ordered (LO) $k$-colouring of a hypergraph assigns to each vertex a colour from the set $\{0,1,\ldots,k-1\}$ in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO $k$-colouring of an LO 2-colourable 3-uniform hypergraph for any constant $k\geq 2$ [STACS'21] but even the case $k=3$ is still open. Nakajima and \v{Z}ivn\'{y} gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with $O^*(\sqrt{n})$ colours [ICALP'22] and an LO colouring with $O^*(\sqrt[3]{n})$ colours [ACM ToCT'23]. We present a simple polynomial-time algorithm that finds an LO colouring with $\log_2(n)$ colours, which is an exponential improvement.

翻译：超图的线性有序k-着色要求为每个顶点分配集合{0,1,...,k-1}中的一种颜色，使得每条超边具有唯一的最大元素。Barto、Batistelli和Berg猜想：对于任意常数k≥2，为可二色线性有序的3一致超图寻找线性有序k-着色问题是NP难的[STACS'21]，但即便k=3的情形仍未解决。Nakajima与Živný给出了多项式时间算法，可在给定可二色线性有序的3一致超图时，分别找到使用O*(\sqrt{n})种颜色[ICALP'22]和O*(\sqrt[3]{n})种颜色[ACM ToCT'23]的线性有序着色方案。我们提出一个简单的多项式时间算法，可找到使用log₂(n)种颜色的线性有序着色方案，这实现了指数级的改进。