𝔖 Bobbio Scriptorium
✦   LIBER   ✦

On the genus of the graph Kn × K2 or the n-prism

✍ Scribed by Gerhard Ringel

Publisher
Elsevier Science
Year
1977
Tongue
English
Weight
591 KB
Volume
20
Category
Article
ISSN
0012-365X

No coin nor oath required. For personal study only.

✦ Synopsis

For all n $ S or 9 (mod 12) the genu3 :lf the granh II, x K, is shown t#c> be equal to the lower hound given by the Euler Formula.

Consider two disjoint copies of a complete graph K,, the first t&h vertices 1,2, . . -, n, the second with vertices l', 2', . . .) n'. Then join vertex i with vertex i' by one edge ft = 1,2. . . ., n). The graph obtained has 2n vertices and n2 edges and can be interpreted as the Cartesian product K,, X Kz and also the n-dimensional generalization of the 3-dimensional three sided prism. The genus y(G) of a graph G is the smallest genus of an orientable surke S s!mch that G can be imbedded into S without crossings of pairs of edges. For instance y(_?C,, x K,) = 0 because K, x K1 is imbeddabk into the sphere. Fig. shows an imheddjng of K, X K2 into the torus.

📜 SIMILAR VOLUMES

The genus of Kn − K2
The genus of Kn − K2
✍ Mark Jungerman 📂 Article 📅 1975 🏛 Elsevier Science 🌐 English ⚖ 324 KB
On the average genus of the random graph
On the average genus of the random graph
✍ Saul Stahl 📂 Article 📅 1995 🏛 John Wiley and Sons 🌐 English ⚖ 640 KB

## Abstract We obtain an upper bound on the expected number of regions in the randomly chosen orientable embedding of a fixed graph. This bound is ised to show that the average genus of the random graph on __v__ vertices is close to its maximum genus. More specifically, it is proven that the differ

On the maximum genus of a graph
On the maximum genus of a graph
✍ E.A Nordhaus; B.M Stewart; A.T White 📂 Article 📅 1971 🏛 Elsevier Science 🌐 English ⚖ 415 KB
Survey of results on the maximum genus o
Survey of results on the maximum genus of a graph
✍ Richard D. Ringeisen 📂 Article 📅 1979 🏛 John Wiley and Sons 🌐 English ⚖ 619 KB

## Abstract Some of the early questions concerning the maximum genus of a graph have now been answered. In this paper we survey the progress made on such problems and present some recent results, outlining proofs for some of the major theorems.