Two product formulas for counting successive vertex orderings

TL;DR

This paper confirms conjectured product formulas for counting successive vertex orderings in specific classes of line graphs derived from complete hypergraphs, extending previous combinatorial enumeration results.

Contribution

It proves two conjectures by Fang et al. on product formulas for counting successive vertex orderings in line graphs of complete hypergraphs.

Findings

Confirmed formulas for line graphs of complete 3-uniform hypergraphs.

Validated product formulas involving binomial coefficients.

Extended combinatorial enumeration techniques for specific graph classes.

Abstract

A vertex ordering of a graph $G$ is a bijection $\pi\colon\{1,\dots,|V(G)|\}\to V(G)$. It is successive if the induced subgraph $G[v_{\pi(1)},\dots,v_{\pi(k)}]$ is connected for each $k$. Lixing Fang, Hao Huang, J\'anos Pach, G\'abor 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.