J. Aust. Math. Soc.  81 (2006), 153-164
Cubic symmetric graphs of order twice an odd prime-power

Yan-Quan Feng
  Department of Mathematics
  Beijing Jiaotong University
  Beijing 100044
  P.R. China
Jin Ho Kwak
  Combinatorial and Computational
  Mathematics Center
  Pohang University of Science and Technology
  Pohang, 790--784

An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p we show that if $p\not=5,7$ then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if $p\not=3$ then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s – 1)-regular subgroup for each $1\leq s\leq 5$. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each $1\leq s\leq 5$ and each prime p, as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.
Download the article in PDF format (size 104 Kb)

Australian Mathematical Publishing Association Inc. ©  Australian MS