In a paper by Cai Heng Li in Bull. London Math. Soc. 33 (2001), it was suggested that ‘non-basic’ $2$-arc-transitive graphs of prime-power order that occur as normal covers of smaller $2$-arc-transitive graphs might be rare and difficult to construct. This note describes some of the background to Li’s suggestion, and gives some examples of small valency, and then goes on to show that in fact there are infinitely many examples of valency $d$, for every integer $d \ge 2$. It is also noted that the hypercubes $Q_n$ for $n\ge 4$ even, together with a $2$-arc-transitive group $G$ of index $2$ in $\mathrm{Aut}(Q_n)$, show that the claims of Corollary 1.2 in the above paper by Li are not quite correct.
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Keywords: graph, $s$-arc-transitive, normal cover
Conder, Marston 1; Potočnik, Primož 2
CC-BY 4.0
@article{ALCO_2025__8_2_469_0,
author = {Conder, Marston and Poto\v{c}nik, Primo\v{z}},
title = {Normal covers of 2-arc-transitive graphs of prime-power order},
journal = {Algebraic Combinatorics},
pages = {469--478},
publisher = {The Combinatorics Consortium},
volume = {8},
number = {2},
year = {2025},
doi = {10.5802/alco.415},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.415/}
}
TY - JOUR AU - Conder, Marston AU - Potočnik, Primož TI - Normal covers of 2-arc-transitive graphs of prime-power order JO - Algebraic Combinatorics PY - 2025 SP - 469 EP - 478 VL - 8 IS - 2 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.415/ DO - 10.5802/alco.415 LA - en ID - ALCO_2025__8_2_469_0 ER -
%0 Journal Article %A Conder, Marston %A Potočnik, Primož %T Normal covers of 2-arc-transitive graphs of prime-power order %J Algebraic Combinatorics %D 2025 %P 469-478 %V 8 %N 2 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.415/ %R 10.5802/alco.415 %G en %F ALCO_2025__8_2_469_0
Conder, Marston; Potočnik, Primož. Normal covers of 2-arc-transitive graphs of prime-power order. Algebraic Combinatorics, Volume 8 (2025) no. 2, pp. 469-478. doi : 10.5802/alco.415. https://alco.centre-mersenne.org/articles/10.5802/alco.415/
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