  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.
