32 research outputs found
Edge-dominating cycles, k-walks and Hamilton prisms in -free graphs
We show that an edge-dominating cycle in a -free graph can be found in
polynomial time; this implies that every 1/(k-1)-tough -free graph admits
a k-walk, and it can be found in polynomial time. For this class of graphs,
this proves a long-standing conjecture due to Jackson and Wormald (1990).
Furthermore, we prove that for any \epsilon>0 every (1+\epsilon)-tough
-free graph is prism-Hamiltonian and give an effective construction of a
Hamiltonian cycle in the corresponding prism, along with few other similar
results.Comment: LaTeX, 8 page
Toughness and hamiltonicity in -trees
We consider toughness conditions that guarantee the existence of a hamiltonian cycle in -trees, a subclass of the class of chordal graphs. By a result of Chen et al.\ 18-tough chordal graphs are hamiltonian, and by a result of Bauer et al.\ there exist nontraceable chordal graphs with toughness arbitrarily close to . It is believed that the best possible value of the toughness guaranteeing hamiltonicity of chordal graphs is less than 18, but the proof of Chen et al.\ indicates that proving a better result could be very complicated. We show that every 1-tough 2-tree on at least three vertices is hamiltonian, a best possible result since 1-toughness is a necessary condition for hamiltonicity. We generalize the result to -trees for : Let be a -tree. If has toughness at least then is hamiltonian. Moreover, we present infinite classes of nonhamiltonian 1-tough -trees for each $k\ge 3
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
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
The spectrum and toughness of regular graphs
In 1995, Brouwer proved that the toughness of a connected -regular graph
is at least , where is the maximum absolute value of
the non-trivial eigenvalues of . Brouwer conjectured that one can improve
this lower bound to and that many graphs (especially graphs
attaining equality in the Hoffman ratio bound for the independence number) have
toughness equal to . In this paper, we improve Brouwer's spectral
bound when the toughness is small and we determine the exact value of the
toughness for many strongly regular graphs attaining equality in the Hoffman
ratio bound such as Lattice graphs, Triangular graphs, complements of
Triangular graphs and complements of point-graphs of generalized quadrangles.
For all these graphs with the exception of the Petersen graph, we confirm
Brouwer's intuition by showing that the toughness equals ,
where is the smallest eigenvalue of the adjacency matrix of the
graph.Comment: 15 pages, 1 figure, accepted to Discrete Applied Mathematics, special
issue dedicated to the "Applications of Graph Spectra in Computer Science"
Conference, Centre de Recerca Matematica (CRM), Bellaterra, Barcelona, June
16-20, 201