Orientably-Regular π\pi-Maps and Regular π\pi-Maps

Given a map with underlying graph $\mathcal{G}$, if the set of prime divisors of $|V(\mathcal{G}|$ is denoted by $\pi$, then we call the map a {\it $\pi$-map}. An orientably-regular (resp. A regular ) $\pi$-map is called {\it solvable} if the group $G^+$ of all orientation-preserving automorphisms (resp. the group $G$ of automorphisms) is solvable; and called {\it normal} if $G^+$ (resp. $G$) contains a normal $\pi$-Hall subgroup. In this paper, it will be proved that orientably-regular $\pi$-maps are solvable and normal if $2\notin \pi$ and regular $\pi$-maps are solvable if $2\notin \pi$ and $G$ has no sections isomorphic to ${\rm PSL}(2,q)$ for some prime power $q$. In particular, it's shown that a regular $\pi$-map with $2\notin \pi$ is normal if and only if $G/O_{2^{'}}(G)$ is isomorphic to a Sylow $2$-group of $G$. Moreover, nonnormal $\pi$-maps will be characterized and some properties and constructions of normal $\pi$-maps will be given in respective sections.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:2201.0430

Nonorientable regular embeddings of graphs of order p2

AbstractA map is called regular if its automorphism group acts regularly on the set of all flags (incident vertex–edge–face triples). An orientable map is called orientably regular if the group of all orientation-preserving automorphisms is regular on the set of all arcs (incident vertex–edge pairs). If an orientably regular map admits also orientation-reversing automorphisms, then it is regular, and is called reflexible. A regular embedding and orientably regular embedding of a graph G are, respectively, 2-cell embeddings of G as a regular map and orientably regular map on some closed surface. In Du et al. (2004) [7], the orientably regular embeddings of graphs of order pq for two primes p and q (p may be equal to q) have been classified, where all the reflexible maps can be easily read from the classification theorem. In [11], Du and Wang (2007) classified the nonorientable regular embeddings of these graphs for p≠q. In this paper, we shall classify the nonorientable regular embeddings of graphs of order p2 where p is a prime so that a complete classification of regular embeddings of graphs of order pq for two primes p and q is obtained. All graphs in this paper are connected and simple