Vertex-Transitive Cubic Graphs of Square-Free Order

Abstract

A classification of connected vertex‐transitive cubic graphs of square‐free order is provided. It is shown that such graphs are well‐characterized metacirculants (including dihedrants, generalized Petersen graphs, Möbius bands), or Tutte's 8‐cage, or graphs arisen from simple groups PSL(2, p).

On edge-transitive cubic graphs of square-free order

“…Suppose that the latter case occurs. By [14], we have A ∼ = D 6m :Z 3 and A + ∼ = Z 3m :Z 3 . On other hand, by the construction of Γ , A + has a subgroup of order 9m which has trivial center.…”

“…Thus an edge-transitive cubic graph is either arc-transitive or semisymmetric. In a recent paper [14], the arc-transitive cubic graphs of square-free order were classified. This motivated us to classify the semisymmetric cubic graphs of square-free order, and thus we can get a complete list of edgetransitive cubic graphs of square-free order.…”

The Two-Arc-Transitive Graphs of Square-Free Order Admitting Alternating or Symmetric Groups

“…The class of graphs of square-free order has been studied in some special cases. Further, see [17,[19][20][21] for the case of order four or more distinct primes. The classification of symmetric graphs with order 3p in [32] and some other graphs with order a product of two distinct primes were classified in [22,27,28].…”

Arc-transitive cubic graphs of order four times an odd square-free integer

“…For cubic graphs, many of the authors classify all symmetric graphs with order of square-free or cube-free order. For square-free order, a classification of arc-regular cubic graph (that is, the full automorphism group acts regularly on its arc set) of all square-free order is presented in [26] and cubic vertex-transitive graphs of all square-free order has been completely classified in [13,27]. For cube-free order, the classifications of cubic arctransitive graphs of order 4p, 4p 2 , 2p 2 and 6p 2 are presented in [9,10,25].…”