A longstanding open question of Archdeacon and Craft asks whether every complete graph has a minimum genus embedding with at most one nontriangular face. We exhibit such an embedding for each complete graph except $K_8$, the complete graph on 8 vertices, and we go on to prove that no such embedding can exist for this graph. Our approach also solves a more general problem, giving a complete characterization of the possible face distributions (i.e. the numbers of faces of each length) realizable by minimum genus embeddings of each complete graph. We also tackle analogous questions for nonorientable and maximum genus embeddings.
Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of the graph is fixed as the complement of some convex region in the plane, and all other vertices are placed at the mass center of their neighbors, then this results in a unique embedding, and this embedding is planar. It also follows fairly quickly that this embedding minimizes the sum of squared edge lengths, conditional on the embedding of the outer face. However, it is not at all clear how to embed this outer face. We consider the minimization problem of embedding this outer face, up to som…
Hartsfield and Ringel constructed orientable quadrangular embeddings of the complete graph $K_n$ for $n\equiv 5 \pmod 8$, and nonorientable ones for $n \ge 9$ and $n\equiv 1 \pmod 4$. These provide minimal quadrangulations of their underlying surfaces. We extend these results to determine, for every complete graph $K_n$, $n \ge 4$, the minimum genus, both orientable and nonorientable, for the surface in which $K_n$ has an embedding with all faces of degree at least $4$, and also for the surface in which $K_n$ has an embedding with all faces of even degree. These last embeddings provide sharpne…
A Classification of Orientable Regular Embeddings of Complete Multipartite Graphs
设$K_{m[n]}$是具有$m$个部分的完全多部图，每个部分包含$n$个顶点. Biggs（1971）\cite{Big1}，James和Jones（1985）\cite{JJ}确定了完全图$K_{m[1]}$的定向正则嵌入.在过去的20年里，一些论文，如杜等。(2007 \cite{DJKNS1,DJKNS2}，Jones et al.（2007，2008）\cite{JNS1,JNS2}，Kwak and Kwon（2005，2008）\cite{KK1,KK2} and Nedela et al.（1997，2002）\cite{NS,NSZ}对完全二部图$K_{2[n]}$的定向正则嵌入做出了贡献，Jones \cite{Jon1}在2010年给出了最终分类。在我们的前一篇文章\cite{ZD}的基础上，本文给出了图$K_{m[n]}$的定向正则嵌入的一个完全分类，并给出了一般情形$m\ge 3$和$n\ge 2$ .
Given a complete simple topological graph $G$, a $k$-face generated by $G$ is the open bounded region enclosed by the edges of a non-self-intersecting $k$-cycle in $G$. Interestingly, there are complete simple topological graphs with the property that every odd face it generates contains the origin. In this paper, we show that every complete $n$-vertex simple topological graph generates at least $Ω(n^{1/3})$ pairwise disjoint 4-faces. As an immediate corollary, every complete simple topological graph on $n$ vertices drawn in the unit square generates a 4-face with area at most $O(n^{-1/3})$. F…
A book embedding of a complete graph is a spatial embedding whose planar projection has the vertices located along a circle, consecutive vertices are connected by arcs of the circle, and the projections of the remaining "interior" edges in the graph are straight line segments between the points on the circle representing the appropriate vertices. A random embedding of a complete graph can be generated by randomly assigning relative heights to these interior edges. We study a family of two-component links that arise as the realizations of pairs of disjoint cycles in these random embeddings of g…
