@article{13165,
abstract = {A graph G=(V, E) is called fully regular if for every independent set I c V, the number of vertices in V\I that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G is called successive if for every i, the first i vertices induce a connected subgraph of G. We give an explicit formula for the number of successive vertex orderings of a fully regular graph.
As an application of our results, we give alternative proofs of two theorems of Stanley and Gao & Peng, determining the number of linear edge orderings of complete graphs and complete bipartite graphs, respectively, with the property that the first i edges induce a connected subgraph.
As another application, we give a simple product formula for the number of linear orderings of the hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every i, the first i hyperedges induce a connected subgraph. We found similar formulas for complete (non-partite) 3-uniform hypergraphs and in another closely related case, but we managed to verify them only when the number of vertices is small.},
author = {Fang, Lixing and Huang, Hao and Pach, János and Tardos, Gábor and Zuo, Junchi},
issn = {1096-0899},
journal = {Journal of Combinatorial Theory. Series A},
number = {10},
publisher = {Elsevier},
title = {{Successive vertex orderings of fully regular graphs}},
doi = {10.1016/j.jcta.2023.105776},
volume = {199},
year = {2023},
}