Saul Stahl

The embeddings of a graph—A survey

Topological graph theory seeks to find answers to the question of how graphs map into surfaces. This paper surveys the information now available about the range of a graph, namely, the set of surfaces on which the graph can be “neatly” embedded. Several other closely related topics, such as irreducible graphs, coloring problems, and crossing numbers, are ignored. As is quite often the case with mathematical theories, this discipline developed in a rather haphazard manner. Many isolated results existed before the practitioners became aware of the fact that they were developing a theory. The turning point occurred in 1968, when Ringel and Youngs completed their proof of the Heawood conjecture. Their proof, in addition to settling an old unsolved problem, also reinforced the significance of the rotation systems. It is the authors belief that these rotation systems, together with the generalized embedding schemes can, and should, become the main tool in all investigations concerning the embeddings of a graph. This survey is written from that point of view. After defining the scope of the area surveyed, this paper proceeds to discuss the significance of the rotation systems and embedding schemes. Several theorems of a general nature are listed. Attention is then focused on the maximum and minimum genera of a graph. Discussion of the first of these is deferred to another survey article by R. Ringeisen to appear in a subsequent issue. The various methods developed by researchers in this area for determining the (minimum) genus are then described. This is followed by a listing of all the theoretical information that is available about the genus parameter. The paper includes two tables that exhibit most of the graphs with known genus.

Region distributions of graph embeddings and Stirling numbers

Abstract It is shown that the distribution of the number of regions r in the random orientable embedding of the graph with one vertex and q loops is approximately proportional to the unsigned Stirling numbers of the first kind s (2 q,r ) where r has different parity from q . This approximation is strong enough to imply that both the limiting mean and variance of this distribution differ from ln 2 q by small known constants. The paper concludes with a result on the unimodality of some recursively defined sequences and also some conjectures regarding region distributions of arbitrary graphs.

Multichromatic numbers, star chromatic numbers and Kneser graphs

We investigate the relation between the multichromatic number (discussed by Stahl and by Hilton, Rado and Scott) and the star chromatic number (introduced by Vince) of a graph. Denoting these by χ* and η*, the work of the above authors shows that χ*(G) = η*(G) if G is bipartite, an odd cycle or a complete graph. We show that χ*(G) ≤ η*(G) for any finite simple graph G. We consider the Kneser graphs , for which χ* = m/n and η*(G)/χ*(G) is unbounded above. We investigate particular classes of these graphs and show that η* = 3 and η* = 4; (n ≥ 1), and η* = m - 2; (m ≥ 4).

Permutation—partition pairs. III: embedding distributions of linear families of graphs

Abstract For any fixed graph H , and H -linear family of graphs is a sequence { G n } n =1 ∞ of graphs in which G n consists of n copies of H that have been linked in a consistent manner so as to form a chain. Generating functions for the region distribution of any such family are found. It is also shown that the minimum genus and the average genus of G n are essentially linear functions of n .

Region distributions of some small diameter graphs

Abstract Let G be a graph with a vertex u such that V ( G ) − { u } induces either a forest or a cycle. It is shown that the region distribution of G is approximately proportional to the Stirling numbers of the first kind.

The multichromatic numbers of some Kneser graphs

Abstract The Kneser graph K(m,n) has the n -subsets of {1,2,…,m} as its vertices, two such vertices being adjacent whenever they are disjoint. The k th multichromatic number of the graph G is the least integer t such that the vertices of G can be assigned k -subsets of {1,2, …, t}, so that adjacent vertices of G receive disjoint sets. The values of X k ( K ( m , n )) are computed for n = 2, 3 and bounded for n ⩾ 4.

ON THE ZEROS OF SOME GENUS POLYNOMIALS

In the genus polynomial of the graph G, the coefficient of x k is the number of distinct embeddings of the graph G on the oriented surface of genus k. It is shown that for several infinite families of graphs all the zeros of the genus polynomial are real and negative. This implies that their coefficients, which constitute the genus distribution of the graph, are log concave and therefore als o unimodal. The geometric distribution of the zeros of some of these polynomials is als o investigated and some new genus polynomials are presented.

A Combinatorial Analog of the Jordan Curve Theorem

The concept of the genus of a pair of permutations is defined in the same manner as was done by Jacques. The integrality of the genus is proven in a new way by applying a technique developed by Walkup for the reduction of products of permutations. These tools are then used to prove an analog of the Jordan Curve Theorem for a pair of permutations whose genus is zero. A Jordan Curve Theorem for plane embeddings of graphs then follows as a corollary.

On the product of certain permutations

A theorem by M. Cohn and A. Lempel, relating the product of certain permutations to the rank of an associated matrix, is restated and reproved. Its relationship to old and recent results in topological graph theory is pointed out.

On the average genus of the random graph

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 difference between these two parameters is bounded by a function that is linear in v. These bounds are obtained in the context of permutation—partition pairs.