An infinite family of simple graphs underlying chiral, orientable reflexible and non-orientable rotary maps
Isabel Hubard, Primož Potočnik, Primož Šparl
TL;DR
The paper identifies an infinite family of simple graphs, specifically the lexicographic products $\,C_n[mK_1]$ with $m\ge3$, $n=sm$, and $s\not\equiv 0\pmod4$, that underlie representatives from all three rotary-map classes: chiral, orientable reflexible, and non-orientable reflexible. It provides explicit constructions of three polytopal maps on each graph: a chiral map of type $\{mn,2m\}$, an orientable reflexible map of type $\{n,2m\}$, and a non-orientable reflexible map of type $\{2n,2m\}$, with precise genus formulas. The work employs detailed automorphism-generator frameworks on $\Gamma$ to realise the maps and proves the absence/presence of involutions to distinguish chirality from reflexibility, and orientability status. This answers a question of Wilson by producing an explicit infinite family of graphs that serve as skeletons for all three map classes, enriching the landscape of regular and semi-regular map-skeletons. The results contribute concrete, algebraically controlled examples with explicit types and genera, and raise further questions about classification and polyhedrality among such graphs.
Abstract
In this paper, we provide the first known infinite family of simple graphs, each of which is the skeleton of a chiral map, a skeleton of a reflexible map on an orientable surfaces, as well as a skeleton of a reflexible map on a non-orientable surface. This family consists of all lexicographic product $C_n[mK_1]$, where $m\ge 3$, $n = sm$, with $s$ an integer not divisible by $4$. This answers a question posed in [S.\ Wilson, Families of regular graphs in regular maps, {\em Journal of Combinatorial Theory, Series B} 85 (2002), 269--289].