Thomas W. Tucker

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.

Genus distributions for bouquets of circles

Abstract The genus distribution of a graph G is defined to be the sequence {gm} such that gm is the number of different imbeddings of G in the closed orientable surface of genus m. A counting formula of D. M. Jackson concerning the cycle structure of permutations is used to derive the genus distribution for any bouquet of circles Bn. It is proved that all these genus distributions for bouquets are strongly unimodal.

Finite groups acting on surfaces and the genus of a group

Abstract The relationship between a finite group action on a closed surface and Cayley graphs for the group embedded in the surface is studied.

Cayley maps

We present a theory of Cayley maps, i.e., embeddings of Cayley graphs into oriented surfaces having the same cyclic rotation of generators around each vertex. These maps have often been used to encode symmetric embeddings of graphs. We also present an algebraic theory of Cayley maps and we apply the theory to determine exactly which regular or edge-transitive tilings of the sphere or plane are Cayley maps or Cayley graphs. Our main goal, however, is to provide the general theory so as to make it easier for others to study Cayley maps.

Regular t-balanced Cayley maps

The concept of a t-balanced Cayley map is a natural generalization of the previously studied notions of balanced and anti-balanced Cayley maps (the terms coined by [J. Siran, M. Skoviera, Groups with sign structure and their antiautomorphisms, Discrete Math. 108 (1992) 189-202. [12]]). We develop a general theory of t-balanced Cayley maps based on the use of skew-morphisms of groups [R. Jajcay, J. Siran, Skew-morphisms of regular Cayley maps, Discrete Math. 244 (1-3) (2002) 167-179], and apply our results to the specific case of regular Cayley maps of abelian groups.

The genera, reflexibility and simplicity of regular maps

This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to g −1, where g is the genus, all orientably-regular maps of genus p+1 for p prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and that orientable surfaces of infinitely many genera carry no reflexible regular map with simple underlying graph. Another consequence is a simpler proof of the Breda–Nedela–Siraň classification of non-orientable regular maps of Euler characteristic −p where p is prime.

Topics in Topological Graph Theory

Preface Foreword Jonathan L. Gross and Thomas W. Tucker Introduction Lowell W. Beineke and Robin J. Wilson 1. Embedding graphs on surfaces Jonathan L. Gross and Thomas W. Tucker 2. Maximum genus Jianer Chen and Yuanqiu Huang 3. Distributions of embeddings Jonathan L. Gross 4. Algorithms and obstructions for embeddings Bojan Mohar 5. Graph minors: generalizing Kuratowskis theorem R. Bruce Richter 6. Colouring graphs on surfaces Joan P. Hutchinson 7. Crossing numbers R. Bruce Richter and G. Salazar 8. Representing graphs and maps Tomaz Pisanski and Arjana Zitnik 9. Enumerating coverings Jin Ho Kwak and Jaeun Lee 10. Symmetric maps Jozef Siran and Thomas W. Tucker 11. The genus of a group Thomas W. Tucker 12. Embeddings and geometries Arthur T. White 13. Embeddings and designs M. J. Grannell and T. S. Griggs 14. Infinite graphs and planar maps Mark E. Watkins 15. Open problems Dan Archdeacon Notes on contributors Index of definitions.

Log-Concavity of Combinations of Sequences and Applications to Genus Distributions

We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed by convolution are log-concave. These conditions involve relations on sequences called \textit{synchronicity} and \textit{ratio-dominance}, and a characterization of some bivariate sequences as \textit{lexicographic}. We are motivated by the 25-year old conjecture that the genus distribution of every graph is log-concave. Although calculating genus distributions is NP-hard, they have been calculated explicitly for many graphs of tractable size, and the three conditions have been observed to occur in the \textit{partitioned genus distributions} of all such graphs. They are used here to prove the log-concavity of the genus distributions of graphs constructed by iterative amalgamation of double-rooted graph fragments whose genus distributions adhere to these conditions, even though it is known that the genus polynomials of some such graphs have imaginary roots. A blend of topological and combinatorial arguments demonstrates that log-concavity is preserved through the iterations.

Straight-ahead walks in Eulerian graphs

A straight-ahead walk in an embedded Eulerian graph G always passes from an edge to the opposite edge in the rotation at the same vertex. A straight-ahead walk is called Eulerian if all the edges of the embedded graph G are traversed in this way starting from an arbitrary edge. An embedding that contains an Eulerian straight-ahead walk is called an Eulerian embedding. In this article, we characterize some properties of Eulerian embeddings of graphs and of embeddings of graphs such that the corresponding medial graph is Eulerian embedded. We prove that in the case of 4-valent planar graphs, the number of straight-ahead walks does not depend on the actual embedding in the plane. Finally, we show that the minimal genus over Eulerian embeddings of a graph can be quite close to the minimal genus over all embeddings.

There is one group of genus two

Abstract The genus of a finite group is the minimum genus over all surfaces containing an imbedded Cayley graph for the group. It is shown that there is exactly one group of genus two.