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Arc‐transitive maps with underlying Rose Window graphs
Journal of Graph Theory ( IF 0.9 ) Pub Date : 2020-07-14 , DOI: 10.1002/jgt.22608
Isabel Hubard 1 , Alejandra Ramos‐Rivera 2 , Primož Šparl 2, 3, 4
Affiliation  

Let ${\cal M}$ be a map with the underlying graph $\Gamma$. The automorphism group $Aut({\cal M})$ induces a natural action on the set of all vertex-edge-face incident triples, called {\em flags} of ${\cal M}$. The map ${\cal M}$ is said to be a {\em $k$-orbit} map if $Aut({\cal M})$ has $k$ orbits on the set of all flags of ${\cal M}$. It is known that there are seven different classes of $2$-orbit maps, with only four of them corresponding to arc-transitive maps, that is maps for which $Aut{\cal M}$ acts arc-transitively on the underlying graph $\Gamma$. The Petrie dual operator links these four classes in two pairs, one of which corresponds to the chiral maps and their Petrie duals. In this paper we focus on the other pair of classes of $2$-orbit arc-transitive maps. We investigate the connection of these maps to consistent cycles of the underlying graph with special emphasis on such maps of smallest possible valence, namely $4$. We then give a complete classification of such maps whose underlying graphs are arc-transitive Rose Window graphs.

中文翻译:

带有底层玫瑰窗图的弧传递图

令 ${\cal M}$ 是一个带有底层图 $\Gamma$ 的映射。自同构群 $Aut({\cal M})$ 在所有顶点-边-面事件三元组的集合上引发一个自然动作,称为 ${\cal M}$ 的 {\em flags}。如果 $Aut({\cal M})$ 在 ${\ 的所有标志的集合上有 $k$ 轨道,则地图 ${\cal M}$ 被称为 {\em $k$-orbit} 地图cal M}$。已知$2$轨道图有七种不同的类别,其中只有四种对应于弧传递图,即$Aut{\cal M}$在底层图$上起弧传递作用的图\伽玛$。Petrie 对偶算子将这四个类分为两对,其中一对对应于手征图及其 Petrie 对偶。在本文中,我们关注另一类 $2$-orbit 弧传递映射。我们研究了这些映射与底层图的一致循环的联系,并特别强调了这种具有最小可能价的映射,即 $4$。然后我们给出了这些地图的完整分类,这些地图的基础图是弧传递玫瑰窗图。
更新日期:2020-07-14
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