Symmetry properties of generalized graph truncations

Abstract

In the generalized truncation construction, one replaces each vertex of a k-regular graph Γ with a copy of a graph Υ of order k. We investigate the symmetry properties of the graphs constructed in this way, especially in connection to the symmetry properties of the graphs Γ and Υ used in the construction. We demonstrate the usefulness of our results by using them to obtain a classification of cubic vertextransitive graphs of girths 3, 4, and 5.

On generalized truncations of complete graphs

“…For the two groups G, G in the above proposition, we shall follow [6] to say that G is the lift of G. The next lemma shows that if Υ ∼ = C n−1 then G is a 2-transitive permutation group on P and the point stabilizer G Vu is either cyclic or dihedral. Lemma 3.3.…”

“…In [6], the symmetry properties of graphs constructed by using the generalized truncations was investigated. In particular, a method for constructing vertex-transitive generalized truncations was proposed (see [6,Construction 4.1 and Theorem 5.1]), and this method was used to construct vertex-transitive generalized truncations of a complete graph K n by a cycle of length n − 1 for some small values of n. The vertex-transitive generalized truncations of a complete graph K n by a graph Υ in context of [6,Theorem 5.1] can be defined as follows.…”

“…Based on the analysis of special generalized truncations of a complete graph K n by a cycle of length n − 1 for some small values of n, the authors of [6] proposed the following problem.…”

“…Problem 1.1 ([6,Problem 5.4]). Classify the special generalized truncations of K n (n ≥ 4) by a cycle of length n − 1.…”

Girth-regular graphs

“…For example, cubic arc-transitive graphs (a graph is called arc-transitive if its automorphism group acts transitively on its arcs, where an arc is an ordered pair of adjacent vertices) and cubic semisymmetric (regular, edgetransitive but not vertex-transitive) graphs of girth up to 9 and 10 have been studied in [5,11] and [6], respectively, and tetravalent edge-transitive graphs of girths 3 and 4 have been considered in [14]. Recently, a classification of all cubic vertex-transitive graphs of girth up to 5 was obtained in [8].…”