Fat Minors in Finitely Presented Groups

Homomorphisms of partial t-trees and edge-colorings of partial 3-trees

Covalence sequences of planar vertex-homogeneous maps

All finite planar graphs are subgraphs of certain planar Cayley graphs of free groups; outerplanar graphs are subgraphs of certain outerplanar Cayley graphs of free groups. It is shown for these particular planar presentations of free groups that special planar is equivalent with outerplanar.KeywordsCayley GraphCyclic PermutationSpecial PlanarOuterplanar GraphLocal FinitenessThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A comprehensive study of percolation in a more general context than the usual $Z^d$ setting is proposed, with particular focus on Cayley graphs, almost transitive graphs, and planar graphs. Results concerning uniqueness of infinite clusters and inequalities for the critical value $p_c$ are given, and a simple planar example exhibiting uniqueness and non-uniqueness for different $p>p_c$ is analyzed. Numerous varied conjectures and problems are proposed, with the hope of setting goals for future research in percolation theory.

We consider strongly regular graphs Γ e (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman–Singleton graph, and the triangular graphs T(q) with q ≡ 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We give a representation of strongly regular semi-Cayley graphs in terms of suitable triples of elements in the group ring ZG. By applying characters of G, this approach allows us to obtain interesting nonexistence results if G is Abelian, in particular, if G is cyclic. For instance, if G is cyclic and n is odd, then all examples must have parameters of the form 2n e 4s2 + 4s + 2, k e 2s2 + s, λ e s2 − 1, and μ e s2s examples are known only for s e 1, 2, and 4 (together with a noncyclic example for s e 3). We also apply our results to obtain new conditions for the existence of strongly regular Cayley graphs on an even number of vertices when the underlying group H has an Abelian normal subgroup of index 2. In particular, we show the nonexistence of nontrivial strongly regular Cayley graphs over dihedral and generalized quaternion groups, as well as over two series of non-Abelian 2-groups. Up to now these have been the only general nonexistence results for strongly regular Cayley graphs over non-Abelian groupss only the first of these cases was previously known.

We study multiplicative residue semigroups that admit planar Cayley graphs. It is proved that the multiplicative semigroup of the residue ring Zn admits a planar Cayley graph if and only if n = 4, 6, 8 or n is a prime number. Examples of minimal systems of generators of multiplicative residue semigroups with respect to some modules and their Cayley graphs are given, illustrating the obtained results.

We find necessary and sufficient conditions for a finitely generated group with more than one end to have a planar Cayley graph.

We study the unitary Cayley graph associated to an arbitrary finite ring, determining precisely its diameter, girth, eigenvalues, vertex and edge connectivity, and vertex and edge chromatic number. We also compute its automorphism group, settling a question of Klotz and Sander. In addition, we classify all planar graphs and perfect graphs within this class.