用于计数连续顶点排序的两个乘积公式
Two product formulas for counting successive vertex orderings
AI总结：
本研究针对完全正则图连续顶点排序计数的两个猜想，通过推导验证了完全3-均匀超图线图与完全二部3-均匀超图线图的排序数均可表示为二项式系数差或比值的乘积形式，证实了已有猜想。
摘要：
图$G$的顶点排序是一个双射$\pi\colon\{1,\dots,|V(G)|\}\to V(G)$。若对每个$k$，诱导子图$G[v_{\pi(1)},\dots,v_{\pi(k)}]$都是连通的，则该排序是连续的。Lixing Fang、Hao Huang、János Pach、Gábor Tardos和Junchi Zuo[J. Comb. Theory A199 (2023), 105776]给出了一类被他们称为“完全正则”的图的连续顶点排序数的计算公式，并猜想这些公式在两种情况下可表示为涉及二项式系数差或比值的特定乘积：当图是完全3-均匀超图的线图$L(K_n^{(3)})$时，或是当图是完全“二部”3-均匀超图的线图$L(K_{m,n}^{(1,2)})$时。在本文中，我们证实了这两个猜想。
英文摘要：
A vertex ordering of a graph $G$ is a bijection $π\colon\{1,\dots,|V(G)|\}\to V(G)$. It is successive if the induced subgraph $G[v_{π(1)},\dots,v_{π(k)}]$ is connected for each $k$. Lixing Fang, Hao Huang, János Pach, Gábor Tardos, and Junchi Zuo [J. Comb. Theory A199 (2023), 105776] gave formulas for counting the number of successive vertex orderings for a class of graphs they called "fully regular," and conjectured that these formulas could be written as certain products involving differences or ratios of binomial coefficients in two cases: When the graph is the line graph $L(K_n^{(3)})$ of the complete $3$-uniform hypergraph, or when it is the line graph $L(K_{m,n}^{(1,2)})$ of a complete "bipartite" $3$-uniform hypergraph. In this paper, we confirm both of these conjectures.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
