Related papers: Hamiltonicity in generalized quasi-dihedral groups

Following a problem posed by Lov\'asz in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that…

In 1982, Durnberger proved that every connected Cayley graph of a finite group with a commutator subgroup of prime order contains a hamiltonian cycle. In this paper, we extend this result to the infinite case. Additionally, we generalize…

A graph $\G$ with a group $H$ of automorphisms acting semiregularly on the vertices with two orbits is called a {\em bi-Cayley graph} over $H$. When $H$ is a normal subgroup of $\Aut(\G)$, we say that $\G$ is {\em normal} with respect to…

A well-known conjecture of Alspach says that every $2k$-regular Cayley graph of an abelian group can be decomposed into Hamiltonian cycles. We consider an analogous question for infinite abelian groups. In this setting one natural analogue…

We construct an infinite family of connected, 2-generated Cayley digraphs Cay(G;a,b) that do not have hamiltonian paths, such that the orders of the generators a and b are arbitrarily large. We also prove that if G is any finite group with…

We study existence of Hamilton cycles in connected Cayley graphs on generalized dihedral groups

Suppose G is a nilpotent, finite group. We show that if {a,b} is any 2-element generating set of G, then the corresponding Cayley digraph Cay(G;a,b) has a hamiltonian path. This implies there is a hamiltonian path in every connected Cayley…

We initiate the study of Hamiltonian cycles up to symmetries of the underlying graph. Our focus lies on the extremal case of Hamiltonian-transitive graphs, i.e., Hamiltonian graphs where, for every pair of Hamiltonian cycles, there is a…

We consider a number of examples of groups together with an infinite conjugation invariant generating set, including: the free group with the generating set of all separable elements; surface groups with the generating set of all…

A circle of an infinite locally finite graph $G$ is the imagine of a homeomorphic mapping of the unit circle $S^1$ in $|G|$, the Freudenthal compactification of $G$. A circle of $G$ is Hamiltonian if it meets every vertex (and then every…

Generalizing a result of Conway, Sloane, and Wilkes for real reflection groups, we show the Cayley graph of an imprimitive complex reflection group with respect to standard generating reflections has a Hamiltonian cycle. This is consistent…

For locally finite infinite graphs the notion of Hamilton cycles can be extended to Hamilton circles, homeomorphic images of $S^1$ in the Freudenthal compactification. In this paper we extend some well-known theorems of the Hamiltonicity of…

For locally finite infinite graphs the notion of Hamilton cycles can be extended to Hamilton circles, homeomorphic images of $S^1$ in the Freudenthal compactification. In this paper we prove of a sufficient condition for the existence of…

We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to…

A graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex-transitive graphs with finitely many ends, and also discuss…

This note is a continuation of the study of the relationship between the geometry of Cayley graphs and the size of its metric-functional boundary. We show that, if there exists a Cayley graph with finitely many Busemann points, then the…

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that its automorphism group ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple…

We show that every finite group occurs as the automorphism group of infinitely many finite (field) extensions of any given Hilbertian field. This extends and unifies previous results of M. Fried and Takahashi on the global field case.

In 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually…

A connected graph is called \emph{geodetic} if there is a unique geodesic between each pair of vertices. In this paper we prove that if a finitely generated group admits a Cayley graph which is geodetic, then the group must be virtually…