Quadrangular embeddings of complete graphs∗
نویسندگان
چکیده
Hartsfield and Ringel proved that a complete graph Kn has an orientable quadrangular embedding if n ≡ 5 (mod 8), and has a nonorientable quadrangular embedding if n ≥ 9 and n ≡ 1 (mod 4). We complete the characterization of complete graphs admitting quadrangular embeddings by showing that Kn has an orientable quadrilateral embedding if n ≡ 0 (mod 8), and has a nonorientable quadrilateral embedding if n ≡ 0 (mod 4). We also determine the order of minimal quadrangulations for some surfaces where the corresponding graphs are not complete.
منابع مشابه
Embeddings of Cubic Halin Graphs: Genus Distributions∗
We derive an O(n)-time algorithm for calculating the genus distribution of a given 3-regular Halin graph G; that is, we calculate the sequence of numbers g0(G), g1(G), g2(G), . . . on the respective orientable surfaces S0, S1, S2, . . . . Key topological features are a quadrangular decomposition of plane Halin graphs and a new recombinant-strands reassembly process that fits pieces together thr...
متن کاملLabeling Subgraph Embeddings and Cordiality of Graphs
Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$. For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...
متن کاملTriangular embeddings of complete graphs
In this paper we describe the generation of all nonorientable triangular embeddings of the complete graphs K12 and K13. (The 59 nonisomorphic orientable triangular embeddings of K12 were found in 1996 by Altshuler, Bokowski and Schuchert, and K13 has no orientable triangular embeddings.) There are 182, 200 nonisomorphic nonorientable triangular embeddings for K12, and 243, 088, 286 for K13. Tri...
متن کاملGraphs embedded into finite projective planes
We study embeddings of graphs in finite projective planes, and present related results for some families of graphs including complete graphs and complete bipartite graphs. We also make connections between embeddings of graphs and the existence of certain substructures in a plane, such as Baer subplanes and arcs.
متن کاملEmbedding Complete Binary Trees into Star Networks
Abs t rac t . Star networks have been proposed as a possible interconnection network for massively parallel computers. In this paper we investigate embeddings of complete binary trees into star networks. Let G and H be two networks represented by simple undirected graphs. An embedding of G into H is an injective mapping f from the vertices of G into the vertices of H. The dilation of the embedd...
متن کامل