2023/12/08 by Trinh, Hiep, Wilson, Trevor M.
#05C60 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2312.05242
The Fibonacci cube Γn is is the graph whose vertices are independent subsets of the path graph of length n, where two such vertices are considered adjacent if they differ by the addition or removal of a single element. Klavžar [1] suggested considering the infinite Fibonacci cube Γ_∞ whose vertices are independent subsets of the one-way infinite path graph with the same adjacency condition. We show that every connected component of Γ_∞ is asymmetric (has no nontrivial automorphism) and no two connected components of Γ_∞ are isomorphic. This follows from our results on a further generalization ΓG where G is a simple, locally finite hypergraph with no isolated vertices.