When dealing with symmetry properties of mathematical objects, one of the fundamental questions is to determine their full automorphism group. In this paper this question is considered in the context of even/odd permutations dichotomy. More precisely: when is it that the existence of automorphisms acting as even permutations on the vertex set of a graph, called even automorphisms, forces the existence of automorphisms that act as odd permutations, called odd automorphisms. As a first step towards resolving the above question, complete information on the existence of odd automorphisms in cubic symmetric graphs is given.
Following Hujdurović et al. (2016), an automorphism of a graph is said to be even/odd if it acts on the vertex set of the graph as an even/odd permutation. In this paper the formula for calculating the number of graphs of order n admitting odd automorphisms and the formula for calculating the number of graphs of order n without odd automorphisms are given together with their asymptotic estimates. The so-called VTO numbers are also defined.
A map is said to be even-closed if all of its automorphisms act like even permutations on the vertex set. In this paper the study of even-closed regular maps is approached by analysing two distinguished families. The first family consists of embeddings of a well-known family of graphs on distinct orientable surfaces, whereas in the second family we consider all graphs having orientable-regular embeddings on a particular surface. In particular, the classification of even-closed orientable-regular embeddings of the complete bipartite graphs and classification of even-closed orientable-regular maps on the torus are given.
Let a group G of automorphisms of a base graph lift along a regular covering projection to a group G' of automorphism of the covering graph. We say that G lifts as a sectional split extension over a G-invariant subset S of vertices of the base graph if there exists a sectional complement to the group of covering transformations, that is, a complement that has an invariant section over S. Sectional complements are characterized from several viewpoints. The connection between the number of sectional complements and invariant sections on one side, and the structure of the split extension itself on the other, is analyzed. In the case when the group of covering transformations is abelian and the covering projection is given implicitly in terms of a voltage assignment on the base graph, an efficient algorithm for testing whether the lifted group has a sectional complement is presented. The method extends to the case when the group of covering transformations is solvable.
Properties of symmetric cubic graphs are described via their rigid cells, which are maximal connected subgraphs fixed pointwise by some involutory automorphism of the graph. This paper completes the description of rigid cells and the corresponding involutions for each of the 17 ‘action types’ of symmetric cubic graphs. This is obtained with the help of odd automorphisms.