11,067 research outputs found
Infinite Hamiltonian paths in Cayley diagraphs of hyperbolic symmetry groups
AbstractThe hyperbolic symmetry groups [p,q], [p,q]+, and [p+, q] have certain natural generating sets. We determine whether or not the corresponding Cayley digraphs have one-way infinite or two-way infinite directed Hamiltonian paths. In addition, the analogous Cayley graphs are shown to have both one-way infinite and two-way infinite Hamiltonian paths
Some local--global phenomena in locally finite graphs
In this paper we present some results for a connected infinite graph with
finite degrees where the properties of balls of small radii guarantee the
existence of some Hamiltonian and connectivity properties of . (For a vertex
of a graph the ball of radius centered at is the subgraph of
induced by the set of vertices whose distance from does not
exceed ). In particular, we prove that if every ball of radius 2 in is
2-connected and satisfies the condition for
each path in , where and are non-adjacent vertices, then
has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017).
Furthermore, we prove that if every ball of radius 1 in satisfies Ore's
condition (1960) then all balls of any radius in are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio
On Hamilton decompositions of infinite circulant graphs
The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph).
Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}
Universal computation by multi-particle quantum walk
A quantum walk is a time-homogeneous quantum-mechanical process on a graph
defined by analogy to classical random walk. The quantum walker is a particle
that moves from a given vertex to adjacent vertices in quantum superposition.
Here we consider a generalization of quantum walk to systems with more than one
walker. A continuous-time multi-particle quantum walk is generated by a
time-independent Hamiltonian with a term corresponding to a single-particle
quantum walk for each particle, along with an interaction term. Multi-particle
quantum walk includes a broad class of interacting many-body systems such as
the Bose-Hubbard model and systems of fermions or distinguishable particles
with nearest-neighbor interactions. We show that multi-particle quantum walk is
capable of universal quantum computation. Since it is also possible to
efficiently simulate a multi-particle quantum walk of the type we consider
using a universal quantum computer, this model exactly captures the power of
quantum computation. In principle our construction could be used as an
architecture for building a scalable quantum computer with no need for
time-dependent control
Irreducible pseudo 2-factor isomorphic cubic bipartite graphs
A bipartite graph is {\em pseudo 2--factor isomorphic} if all its 2--factors
have the same parity of number of circuits. In \cite{ADJLS} we proved that the
only essentially 4--edge-connected pseudo 2--factor isomorphic cubic bipartite
graph of girth 4 is , and conjectured \cite[Conjecture 3.6]{ADJLS}
that the only essentially 4--edge-connected cubic bipartite graphs are
, the Heawood graph and the Pappus graph.
There exists a characterization of symmetric configurations %{\bf
decide notation and how to use it in the rest of the paper} due to Martinetti
(1886) in which all symmetric configurations can be obtained from an
infinite set of so called {\em irreducible} configurations \cite{VM}. The list
of irreducible configurations has been completed by Boben \cite{B} in terms of
their {\em irreducible Levi graphs}.
In this paper we characterize irreducible pseudo 2--factor isomorphic cubic
bipartite graphs proving that the only pseudo 2--factor isomorphic irreducible
Levi graphs are the Heawood and Pappus graphs. Moreover, the obtained
characterization allows us to partially prove the above Conjecture
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