347 research outputs found
The generalized 3-connectivity of Cartesian product graphs
The generalized connectivity of a graph, which was introduced recently by
Chartrand et al., is a generalization of the concept of vertex connectivity.
Let be a nonempty set of vertices of , a collection
of trees in is said to be internally disjoint trees
connecting if and for
any pair of distinct integers , where . For an integer
with , the -connectivity of is the
greatest positive integer for which contains at least internally
disjoint trees connecting for any set of vertices of .
Obviously, is the connectivity of . Sabidussi showed
that for any two connected graphs
and . In this paper, we first study the 3-connectivity of the Cartesian
product of a graph and a tree , and show that if
, then ;
if , then .
Furthermore, for any two connected graphs and with
, if , then ; if , then
. Our result could be seen as
a generalization of Sabidussi's result. Moreover, all the bounds are sharp.Comment: 17 page
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
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