401 research outputs found
Further topics in connectivity
Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered.
For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version
Oriented paths in n-chromatic digraphs
In this thesis, we try to treat the problem of oriented paths in n-chromatic
digraphs. We first treat the case of antidirected paths in 5-chromatic
digraphs, where we explain El-Sahili's theorem and provide an elementary and
shorter proof of it. We then treat the case of paths with two blocks in
n-chromatic digraphs with n greater than 4, where we explain the two different
approaches of Addario-Berry et al. and of El-Sahili. We indicate a mistake in
Addario-Berry et al.'s proof and provide a correction for it.Comment: 25 pages, Master thesis in Graph Theory at the Lebanese Universit
Superconnectivity of Networks Modeled by the Strong Product of Graphs
Maximal connectivity and superconnectivity in a network are two important
features of its reliability. In this paper, using graph terminology, we first
give a lower bound for the vertex connectivity of the strong product of two
networks and then we prove that the resulting structure is more reliable
than its generators. Namely, sufficient conditions for a strong product of two
networks to be maximally connected and superconnected are given.Ministerio de Economía y Competitividad MTM2014-60127-
On the size of maximally non-hamiltonian digraphs
A graph is called maximally non-hamiltonian if it is non-hamiltonian, yet for any two non-adjacent vertices there exists a hamiltonian path between them. In this paper, we naturally extend the concept to directed graphs and bound their size from below and above. Our results on the lower bound constitute our main contribution, while the upper bound can be obtained using a result of Lewin, but we give here a different proof. We describe digraphs attaining the upper bound, but whether our lower bound can be improved remains open
The mincut graph of a graph
In this paper we introduce an intersection graph of a graph , with vertex
set the minimum edge-cuts of . We find the minimum cut-set graphs of some
well-known families of graphs and show that every graph is a minimum cut-set
graph, henceforth called a \emph{mincut graph}. Furthermore, we show that
non-isomorphic graphs can have isomorphic mincut graphs and ask the question
whether there are sufficient conditions for two graphs to have isomorphic
mincut graphs. We introduce the -intersection number of a graph , the
smallest number of elements we need in in order to have a family of subsets, such that for each subset. Finally we
investigate the effect of certain graph operations on the mincut graphs of some
families of graphs
Maximally Edge-Connected Realizations and Kundu's -factor Theorem
A simple graph with edge-connectivity and minimum degree
is maximally edge connected if . In 1964,
given a non-increasing degree sequence , Jack Edmonds
showed that there is a realization of that is -edge-connected if
and only if with when .
We strengthen Edmonds's result by showing that given a realization of
if is a spanning subgraph of with
such that when , then there is a
maximally edge-connected realization of with as a
subgraph. Our theorem tells us that there is a maximally edge-connected
realization of that differs from by at most edges. For
, if has a spanning forest with components,
then our theorem says there is a maximally edge-connected realization that
differs from by at most edges. As an application we combine our
work with Kundu's -factor Theorem to find maximally edge-connected
realizations with a -factor for and
present a partial result to a conjecture that strengthens the regular case of
Kundu's -factor theorem.Comment: 13 pages, 1 figur
- …