522 research outputs found
Spanning Trees and Spanning Eulerian Subgraphs with Small Degrees. II
Let be a connected graph with and with the spanning
forest . Let be a real number and let be a real function. In this paper, we show that if for all
, , then has a spanning tree
containing such that for each vertex , , where
denotes the number of components of and denotes the
number of edges of with both ends in . This is an improvement of several
results and the condition is best possible. Next, we also investigate an
extension for this result and deduce that every -edge-connected graph
has a spanning subgraph containing edge-disjoint spanning trees such
that for each vertex , , where ; also if contains
edge-disjoint spanning trees, then can be found such that for each vertex
, , where .
Finally, we show that strongly -tough graphs, including -tough
graphs of order at least three, have spanning Eulerian subgraphs whose degrees
lie in the set . In addition, we show that every -tough graph has
spanning closed walk meeting each vertex at most times and prove a
long-standing conjecture due to Jackson and Wormald (1990).Comment: 46 pages, Keywords: Spanning tree; spanning Eulerian; spanning closed
walk; connected factor; toughness; total exces
Factors and Connected Factors in Tough Graphs with High Isolated Toughness
In this paper, we show that every -tough graph with order and isolated
toughness at least has a factor whose degrees are , except for at most
one vertex with degree . Using this result, we conclude that every
-tough graph with order and isolated toughness at least has a
connected factor whose degrees lie in the set , where .
Also, we show that this factor can be found -tree-connected, when is a
-tough graph with order and isolated toughness at least ,
where and . Next, we prove that
every -tough graph of order at least with high enough
isolated toughness admits an -tree-connected factor with maximum degree at
most . From this result, we derive that every -tough graph
of order at least three with high enough isolated toughness has a spanning
Eulerian subgraph whose degrees lie in the set . In addition, we
provide a family of -tough graphs with high enough isolated toughness
having no connected even factors with bounded maximum degree
Approximating ATSP by Relaxing Connectivity
The standard LP relaxation of the asymmetric traveling salesman problem has
been conjectured to have a constant integrality gap in the metric case. We
prove this conjecture when restricted to shortest path metrics of node-weighted
digraphs. Our arguments are constructive and give a constant factor
approximation algorithm for these metrics. We remark that the considered case
is more general than the directed analog of the special case of the symmetric
traveling salesman problem for which there were recent improvements on
Christofides' algorithm.
The main idea of our approach is to first consider an easier problem obtained
by significantly relaxing the general connectivity requirements into local
connectivity conditions. For this relaxed problem, it is quite easy to give an
algorithm with a guarantee of 3 on node-weighted shortest path metrics. More
surprisingly, we then show that any algorithm (irrespective of the metric) for
the relaxed problem can be turned into an algorithm for the asymmetric
traveling salesman problem by only losing a small constant factor in the
performance guarantee. This leaves open the intriguing task of designing a
"good" algorithm for the relaxed problem on general metrics.Comment: 25 pages, 2 figures, fixed some typos in previous versio
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