4,059 research outputs found
Properties of minimally -tough graphs
A graph is minimally -tough if the toughness of is and the
deletion of any edge from decreases the toughness. Kriesell conjectured
that for every minimally -tough graph the minimum degree . We
show that in every minimally -tough graph . We
also prove that every minimally -tough claw-free graph is a cycle. On the
other hand, we show that for every any graph can be embedded
as an induced subgraph into a minimally -tough graph
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
Minimally toughness in special graph classes
Let be a positive real number. A graph is called -tough, if the
removal of any cutset leaves at most components. The toughness of a
graph is the largest for which the graph is -tough. A graph is minimally
-tough, if the toughness of the graph is and the deletion of any edge
from the graph decreases the toughness. In this paper we investigate the
minimum degree and the recognizability of minimally -tough graphs in the
class of chordal graphs, split graphs, claw-free graphs and -free graphs
Conditions for minimally tough graphs
Katona, Solt\'esz, and Varga showed that no induced subgraph can be excluded
from the class of minimally tough graphs. In this paper, we consider the
opposite question, namely which induced subgraphs, if any, must necessarily be
present in each minimally -tough graph.
Katona and Varga showed that for any rational number , every
minimally -tough graph contains a hole. We complement this result by showing
that for any rational number , every minimally -tough graph must
contain either a hole or an induced subgraph isomorphic to the -sun for some
integer .
We also show that for any rational number , every minimally
-tough graph must contain either an induced -cycle, an induced -cycle,
or two independent edges as an induced subgraph
On the minimum degree of minimally -tough, claw-free graphs
A graph is minimally -tough if the toughness of is and
deletion of any edge from decreases its toughness. Katona et al.
conjectured that the minimum degree of any minimally -tough graph is and proved that the minimum degree of minimally -tough and -tough, claw-free graphs is 1 and 2, respectively. We have
show that every minimally -tough, claw-free graph has a vertex of degree
of . In this paper, we give an upper bound on the minimum degree of
minimally -tough, claw-free graphs for
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