Abstract A map is called edge-regular if it is edge-transitive but not arc-transitive. In this paper, we show that a complete graph K n {K}_{n} has an orientable edge-regular embedding if and only if n = p d > 3 n={p}^{d}\gt 3 with p an odd prime such that p d ≡ 3 {p}^{d}\equiv 3 ( mod 4 ) (\mathrm{mod}\hspace{.25em}4) . Furthermore, K p d {K}_{{p}^{d}} has p d − 3 4 d ϕ ( p d − 1 2 ) \tfrac{{p}^{d}-3}{4d}\hspace{0.25em}\phi \left(\tfrac{{p}^{d}-1}{2}\right) non-isomorphic orientable edge-regular embeddings.

中文翻译：

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边正则完全图

摘要 如果地图是边可传递的而不是弧可传递的，则称其为边正则图。在本文中，我们证明了一个完整的图 K n {K}_{n} 具有可定向边正则嵌入当且仅当 n = pd > 3 n={p}^{d}\gt 3 且 p an奇素数使得 pd ≡ 3 {p}^{d}\equiv 3 ( mod 4 ) (\mathrm{mod}\hspace{.25em}4) 。此外，K pd {K}_{{p}^{d}} 有 pd − 3 4 d ϕ ( pd − 1 2 ) \tfrac{{p}^{d}-3}{4d}\hspace{0.25 em}\phi \left(\tfrac{{p}^{d}-1}{2}\right) 非同构定向边缘规则嵌入。