Paper Info
Title
Edge-transitive bi-Cayley graphs
Abstract
A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a bi-Cayley graph over H. This generalisation of a Cayley graph gives a class of graphs that includes many important examples such as the Petersen graph, the Gray graph and the Hoffman-Singleton graph. A bi-Cayley graph Γ over a group H is called normal if H is normal in the full automorphism group of Γ, and almost-normal edge-transitive if the normaliser of H in the full automorphism group of Γ is transitive on the edges of Γ. In this paper, we give a characterisation of almost-normal edge-transitive bi-Cayley graphs, and in particular, we give a detailed description of 2-arc-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic p-groups. We find that under certain conditions, ‘almost-normal edge-transitive’ is the same as ‘normal’ for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most 6. Also we answer some open questions from the literature about these and related classes of graphs, posed by Li (in Proc. Amer. Math. Soc. 133 (2005)), Marušič and Potočnik (in Eur. J. Comb. 22 (2001)) and Marušič and Šparl (in J. Algebr. Comb. 28 (2008)), and we correct a major error in a recent paper by Li, Song and Wang (in J. Comb. Theory Ser. A 120 (2013)).
Year
DOI
Venue
2020
10.1016/j.jctb.2020.05.006
Journal of Combinatorial Theory, Series B
Keywords
DocType
Volume
Bi-Cayley graph,Edge-transitive,Semisymmetric,Half-arc-transitive
Journal
145
ISSN
Citations 
PageRank 
0095-8956
0
0.34
References 
Authors
0
4
Name
Order
Citations
PageRank
Marston D. E. Conder123334.35
Jin-Xin Zhou215625.22
Yan-quan Feng335041.80
Mi-Mi Zhang400.68