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]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with $O^*(\sqrt[5]{n})$ colours. We present two simple polynomial-time algorithms that find an LO colouring with $O(\log_2(n))$ colours, which is an exponential improvement.

翻译：超图的线性序（LO）$k$-着色为每个顶点分配来自集合$\{0,1,\ldots,k-1\}$中的一种颜色，使得每个超边具有唯一的最大元素。Barto、Batistelli和Berg推测：对于任意常数$k\geq 2$，对可LO 2着色的3一致超图寻找LO $k$-着色是NP困难的[STACS'21]，但即使$k=3$的情况仍悬而未决。Nakajima与Živný提出了多项式时间算法，给定可LO 2着色的3一致超图，可分别用$O^*(\sqrt{n})$种颜色[ICALP'22]和$O^*(\sqrt[3]{n})$种颜色[ACM ToCT'23]完成LO着色。近期，Louis、Newman和Ray给出基于半定规划（SDP）的算法，使用$O^*(\sqrt[5]{n})$种颜色。我们提出两个简单的多项式时间算法，能使用$O(\log_2(n))$种颜色完成LO着色，实现了指数级改进。