Odd automorphisms in vertex-transitive graphs (Q2827794)

An automorphism of a graph is even/odd if it acts on the vertex set of the graph as an even/odd permutation. A graph is even closed when all of its automorphisms are even. The central problem of this article is the determination of which vertex-transitive graphs admit odd automorphisms. Particular attention is given to Cayley graphs, especially to circulants, which are Cayley graphs of cyclic groups. For example, given a circulant \(X\) of order \(n\), if \(n\) is even or \(n\equiv3\pmod4\), then \(X\) admits odd automorphisms, but not conversely. A Cayley graph \(X=\mathrm{Cay}(G,S)\) is said to be normal if the subgroup of \(\mathrm{Aut}(X)\) consisting of all left-multiplications \(x\mapsto gx\) for \(g\in G\) is a normal subgroup of \(\mathrm{Aut}(X)\). In a result too technical to quote here, normal, arc-transitive, even closed circulants of order \(n\) are characterized in terms of the prime decomposition of \(n\). The authors pose the problem of classifying cubic vertex-transitive graphs whose vertex-stabilizers are 2-groups that admit odd automorphisms.