具有指定面的稠密 Euler 图的最大亏格嵌入
Maximum genus embeddings of dense eulerian graphs with specified faces
AI总结：
本文研究了稠密 Euler 图或有向图在可定向曲面中的最大亏格嵌入问题，提出了一个密度条件，证明在满足奇偶性要求下，图可胞腔嵌入并恰好形成两个由 Euler 回路界定的面（可预先指定），且涵盖竞赛图等特例。
摘要：
我们给出了一个密度条件，使得在满足必要的奇偶性条件的前提下，Euler 图或有向图可以胞腔嵌入到一个可定向曲面中，使其恰好有两个面，每个面由一条 Euler 回路界定，其中一个面可以预先指定。更一般地，假设一个 n 顶点 Euler 有向图（允许自环和多重弧）的每个顶点至少有 (4n+2)/5 个邻居，并指定弧集分解为不相交的有向回路（闭合迹）。我们证明这样的有向图存在一个可定向嵌入，其中给定的回路为面边界游走，且恰好还有一个或两个其他面。相对于给定回路作为面边界游走，该嵌入具有最大亏格。当仅存在一个其他面时，它必然由一条 Euler 回路界定。因此，如果顶点数和边数具有相同的奇偶性，一个具有给定有向 Euler 回路 T 的足够稠密有向图 D 存在一个恰好有两个面的可定向嵌入，每个面均由 Euler 回路界定，其中之一为 T。有向图的这些结果直接给出了图的相关推论。主要定理涵盖了文献中的若干特殊情况，例如有向图为竞赛图的情况。
英文摘要：
We give a density condition for when, subject to a necessary parity condition, an eulerian graph or digraph may be cellularly embedded in an orientable surface so that it has exactly two faces, each bounded by an euler circuit, one of which may be specified in advance. More generally, suppose that every vertex in an $n$-vertex eulerian digraph (loops and multiple arcs allowed) has at least $(4n+2)/5$ neighbors, and specify any decomposition of the arcs into disjoint directed circuits (closed trails). We show that such a digraph has an orientable embedding in which the given circuits are facial walks and there are exactly one or two other faces. This embedding then has maximum genus relative to the given circuits being facial walks. When there is only one other face, it is necessarily bounded by an euler circuit. Consequently, if the numbers of vertices and edges have the same parity, a sufficiently dense digraph $D$ with a given directed euler circuit $T$ has an orientable embedding with exactly two faces, each bounded by an euler circuit, one of which is $T$. These results for digraphs give analogous results for graphs as immediate corollaries. The main theorem encompasses several special cases in the literature, such as when the digraph is a tournament.
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