65 research outputs found
Hamilton cycles, minimum degree and bipartite holes
We present a tight extremal threshold for the existence of Hamilton cycles in
graphs with large minimum degree and without a large ``bipartite hole`` (two
disjoint sets of vertices with no edges between them). This result extends
Dirac's classical theorem, and is related to a theorem of Chv\'atal and
Erd\H{o}s.
In detail, an -bipartite-hole in a graph consists of two disjoint
sets of vertices and with and such that there are no
edges between and ; and is the maximum integer
such that contains an -bipartite-hole for every pair of
non-negative integers and with . Our central theorem is that
a graph with at least vertices is Hamiltonian if its minimum degree is
at least .
From the proof we obtain a polynomial time algorithm that either finds a
Hamilton cycle or a large bipartite hole. The theorem also yields a condition
for the existence of edge-disjoint Hamilton cycles. We see that for dense
random graphs , the probability of failing to contain many
edge-disjoint Hamilton cycles is . Finally, we discuss
the complexity of calculating and approximating
Partitioning a graph into highly connected subgraphs
Given , a -proper partition of a graph is a partition
of such that each part of induces a
-connected subgraph of . We prove that if is a graph of order
such that , then has a -proper partition with at
most parts. The bounds on the number of parts and the minimum
degree are both best possible. We then prove that If is a graph of order
with minimum degree , where
, then has a -proper partition into at most
parts. This improves a result of Ferrara, Magnant and
Wenger [Conditions for Families of Disjoint -connected Subgraphs in a Graph,
Discrete Math. 313 (2013), 760--764] and both the degree condition and the
number of parts are best possible up to the constant
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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