8 research outputs found
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 vertex-transitive graphs with a unique hamiltonian cycle
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 some examples with infinitely many ends. In particular, we show each
nonabelian free group has a Cayley graph of degree that has a
unique hamiltonian circle. (A weaker statement had been conjectured by A.
Georgakopoulos.) Furthermore, we prove that these Cayley graphs of are
outerplanar