Jonathan L. Gross

Generating all graph coverings by permutation voltage assignments

Abstract This paper introduces the permutation voltage graph construction, which is a generalization of Grosss ordinary voltage graph construction. It is shown that every covering of a given graph arises from some permutation voltage assignment in a symmetric group and that every regular covering (in the topological sense) arises from some ordinary voltage assignment. These results are related to graph imbedding theory. It is demonstrated that the relationship of permutation voltages to ordinary voltages is analogous to the relationship of Schreier coset graphs to Cayley graphs.

Hierarchy for Imbedding-Distribution Invariants of a Graph

Most existing papers about graph imbeddings are concerned with the determination of minimum genus, and various others have been devoted to maximum genus or to highly symmetric imbeddings of special graphs. An entirely different viewpoint is now presented in which one seeks distributional information about the huge family of all cellular imbeddings of a graph into all closed surfaces, instead of focusing on just one imbedding or on the existence of imbeddings into just one surface. The distribution of imbeddings admits a hierarchically ordered class of computable invariants, each of which partitions the set of all graphs into much finer subcategories than the subcategories corresponding to minimum genus or to any other single imbedding surface. Quite low in this hierarchy are invariants such as the average genus, taken over all cellular imbeddings, and the average region size, where “region size” means the number of edge traversals required to complete a tour of a region boundary. Further up in the hierarchy is the multiset of duals of a graph. At an intermediate level are the “imbedding polynomials.” The hierarchy is explored, and several specific calculations of the values of some of the invariants are provided. The main results are concerned with the amount of work needed to derive one invariant from another, when possible, and with principles for computing the algebraic effect of adding an edge or of otherwise combining two graphs.

Genus distributions for two classes of graphs

Abstract The set of orientable imbeddings of a graph can be partitioned according to the genus of the imbedding surfaces. A genus-respecting breakdown of the number of orientable imbeddings is obtained for every graph in each of two infinite classes. It is proved that the genus distribution of any member of either class is strongly unimodal. These are the first two infinite classes of graphs for which such calculations have been achieved, except for a few classes, such as trees and cycles, whose members have all their cellular orientable imbeddings in the sphere.

The topological theory of current graphs

Abstract W. Gustins introduction of combinatorial current graphs as a device for obtaining orientable imbeddings of Cayley “color” graphs was fundamental to the solution of the Heawood map-coloring problem by G. Ringel, J. W. T. Youngs, C. M. Terry, and L. R. Welch. The topological current graphs of this paper lead to a construction that generalizes the method of Gustin and its augmentation to “vortex” graphs by Youngs, extending the scope of current graph theory from Cayley graphs alone to the much larger class of graphs that are covering spaces.

Finding a maximum-genus graph imbedding

The computational complexity of constructing the imbeddings of a given graph into surfaces of different genus is not well understood. In this paper, topological methods and a reduction to linear matroid parity are used to develop a polynomial-time algorithm to find a maximum-genus cellular imbedding. This seems to be the first imbedding algorithm for which the running time is not exponential in the genus of the imbedding surface.

Every connected regular graph of even degree is a Schreier coset graph

Abstract Using Petersens theorem, that every regular graph of even degree is 2-factorable, it is proved that every connected regular graph of even degree is isomorphic to a Schreier coset graph. The method used is a special application of the permutation voltage graph construction developed by the author and Tucker. This work is related to graph imbedding theory, because a Schreier coset graph is a covering space of a bouquet of circles.

Overlap matrices and total imbedding distributions

Abstract The concept of genus distribution of graphs is generalized to include nonorientable imbeddings. Explicit computations of the total imbedding distributions for several interesting graph classes are given. These computations are an illustration of the power of a theorem by Mohar that relates topological types of imbedding surfaces to ranks of the corresponding overlap matrices.

Maximum genus and connectivity

Abstract It is shown that ⌈β(G)/3⌉ is the tight lower bound on the maximum genus γM(G) of 2-edge-connected simplicial graphs, where β(G) is the cycle rank of the graph G. Also, a systematic method is developed to construct 3-vertex-connected simplicial graphs G satisfying the equality γM(G) = ⌈β(G)/3⌉. These two results combine with previously known results to yield a complete picture of the tight lower bounds on the maximum genus of simplicial graphs.

Genus Distributions of Cubic Outerplanar Graphs

We present a quadratic-time algorithm for computing the genus distribution of any 3-regular outerplanar graph. Although recursions and some formulas for genus distributions have previously been calculated for bouquets and for various kinds of ladders and other special families of graphs, cubic outerplanar graphs now emerge as the most general family of graphs whose genus distributions are known to be computable in polynomial time. The key algorithmic features are the syntheses of the given outerplanar graph by a sequence of edge-amalgamations of some of its subgraphs, in the order corresponding to the post-order traversal of a plane tree that we call the inner tree, and the coordination of that synthesis with just-in-time root-splitting.

A tight lower bound on the maximum genus of a simplicial graph

Abstract It is proved that every connected simplicial graph with minimum valence at least three has maximum genus at least one-quarter of its cycle rank. This follows from the technical result that every 3-regular simplicial graph except K 4 has a Xuong co-tree whose odd components have only one edge each. It is proved, furthermore, that this lower bound is tight. However, examples are used to illustrate that it does not apply to non-simplicial graphs. This result on maximum genus leads to several immediate consequences for average genus.