Abstract
A circulant of order $n$ is a Cayley graph for the cyclic group $\mathbb{Z}_n$, and as such, admits a transitive action of $\mathbb{Z}_n$ on its vertices. This paper concerns 2-cell embeddings of connected circulants on closed orientable surfaces. Embeddings on the sphere (the planar case) were classified by Heuberger (2003), and by a theorem of Thomassen (1991), there are only finitely many vertex-transitive graphs with minimum genus $g$, for any given integer $g \ge 3$. Here we completely determine all connected circulants with minimum genus 1 or 2; this corrects and extends an attempted classification of all toroidal circulants by Costa, Strapasson, Alves and Carlos (2010).
Highlights
A circulant is a Cayley graph for a cyclic group
We completely determine all connected circulants with minimum genus 1 or 2; this corrects and extends an attempted classification of all toroidal circulants by Costa, Strapasson, Alves and Carlos (2010)
If n is any positive integer and X is any subset of Zn \ {0} = {1, 2, . . . , n − 1}, the circulant Cn(X) is the undirected simple graph of order n with vertex-set Zn = {0, . . . , n − 1} and edge-set {{i, i + a} : i ∈ Zn, a ∈ X}
Summary
A circulant is a Cayley graph for a cyclic group. To be more precise, if n is any positive integer and X is any subset of Zn \ {0} = {1, 2, . . . , n − 1}, the circulant Cn(X) is the undirected simple graph of order n with vertex-set Zn = {0, . . . , n − 1} and edge-set {{i, i + a} : i ∈ Zn, a ∈ X}. All of them are upper-embeddable, which means that they all have an embedding on an orientable surface (of maximum possible genus), with just one or two faces This is trivial for valency 2 (simple cycles), and for valency greater than 3 it follows from a more general theorem of Skoviera and Nedela about upper-embeddability of connected finite Cayley graphs; see [11, Proposition 7]. For valency 3, it follows from Theorem 5 in [11]), since up to isomorphism the only 3-valent connected circulants of girth 3 are C4(1, 2) ∼= K4 and the triangular prism graph C6(2, 3), both of which are upper-embeddable. M = 2 and n is even and either a2 = ±2a1 or a1 = ±2a2 in Zn. Equivalently, every planar connected circulant graph is isomorphic to either the simple n-cycle.
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