8 research outputs found
Arc-disjoint strong spanning subdigraphs in compositions and products of digraphs
A digraph has a good decomposition if has two disjoint sets
and such that both and are strong. Let be a
digraph with vertices and let be digraphs
such that has vertices Then the
composition is a digraph with vertex set
and arc set
For digraph compositions , we obtain sufficient
conditions for to have a good decomposition and a characterization of
with a good decomposition when is a strong semicomplete digraph and each
is an arbitrary digraph with at least two vertices.
For digraph products, we prove the following: (a) if is an integer
and is a strong digraph which has a collection of arc-disjoint cycles
covering all vertices, then the Cartesian product digraph (the
th powers with respect to Cartesian product) has a good decomposition; (b)
for any strong digraphs , the strong product has a good
decomposition
Proximity and Remoteness in Directed and Undirected Graphs
Let be a strongly connected digraph. The average distance
of a vertex of is the arithmetic mean of the
distances from to all other vertices of . The remoteness and
proximity of are the maximum and the minimum of the average
distances of the vertices of , respectively. We obtain sharp upper and lower
bounds on and as a function of the order of and
describe the extreme digraphs for all the bounds. We also obtain such bounds
for strong tournaments. We show that for a strong tournament , we have
if and only if is regular. Due to this result, one may
conjecture that every strong digraph with is regular. We
present an infinite family of non-regular strong digraphs such that
We describe such a family for undirected graphs as well
Proper orientation number of triangle-free bridgeless outerplanar graphs
An orientation of is a digraph obtained from by replacing each edge
by exactly one of two possible arcs with the same endpoints. We call an
orientation \emph{proper} if neighbouring vertices have different in-degrees.
The proper orientation number of a graph , denoted by , is
the minimum maximum in-degree of a proper orientation of G. Araujo et al.
(Theor. Comput. Sci. 639 (2016) 14--25) asked whether there is a constant
such that for every outerplanar graph and showed that
for every cactus We prove that
if is a triangle-free -connected outerplanar graph and
if is a triangle-free bridgeless outerplanar graph
On Seymour's and Sullivan's Second Neighbourhood Conjectures
For a vertex of a digraph, (, resp.) is the number of
vertices at distance 1 from (to, resp.) and is the number of
vertices at distance 2 from . In 1995, Seymour conjectured that for any
oriented graph there exists a vertex such that .
In 2006, Sullivan conjectured that there exists a vertex in such that
. We give a sufficient condition in terms of the number
of transitive triangles for an oriented graph to satisfy Sullivan's conjecture.
In particular, this implies that Sullivan's conjecture holds for all
orientations of planar graphs and of triangle-free graphs. An oriented graph
is an oriented split graph if the vertices of can be partitioned into
vertex sets and such that is an independent set and induces a
tournament. We also show that the two conjectures hold for some families of
oriented split graphs, in particular, when induces a regular or an almost
regular tournament.Comment: 14 pages, 1 figure
k-Ary spanning trees contained in tournaments
A rooted tree is called a -ary tree, if all non-leaf vertices have exactly
children, except possibly one non-leaf vertex has at most children.
Denote by the minimum integer such that every tournament of order at
least contains a -ary spanning tree. It is well-known that every
tournament contains a Hamiltonian path, which implies that . Lu et al.
[J. Graph Theory {\bf 30}(1999) 167--176] proved the existence of , and
showed that and . The exact values of remain unknown
for . A result of Erd\H{o}s on the domination number of tournaments
implies . In this paper, we prove that and
.Comment: 11 pages, to appear in Discrete Applied Mathematic
Kings in Multipartite Hypertournaments
In his paper "Kings in Bipartite Hypertournaments" (Graphs Combinatorics
35, 2019), Petrovic stated two conjectures on 4-kings in multipartite
hypertournaments. We prove one of these conjectures and give counterexamples
for the other