65 research outputs found
New Bounds for the Dichromatic Number of a Digraph
The chromatic number of a graph , denoted by , is the minimum
such that admits a -coloring of its vertex set in such a way that each
color class is an independent set (a set of pairwise non-adjacent vertices).
The dichromatic number of a digraph , denoted by , is the minimum
such that admits a -coloring of its vertex set in such a way that
each color class is acyclic.
In 1976, Bondy proved that the chromatic number of a digraph is at most
its circumference, the length of a longest cycle.
Given a digraph , we will construct three different graphs whose chromatic
numbers bound .
Moreover, we prove: i) for integers , and with and for each , that if all
cycles in have length modulo for some ,
then ; ii) if has girth and there are integers
and , with such that contains no cycle of length
modulo for each , then ;
iii) if has girth , the length of a shortest cycle, and circumference
, then , which improves,
substantially, the bound proposed by Bondy. Our results show that if we have
more information about the lengths of cycles in a digraph, then we can improve
the bounds for the dichromatic number known until now.Comment: 14 page
Independent sets and non-augmentable paths in generalizations of tournaments
AbstractWe study different classes of digraphs, which are generalizations of tournaments, to have the property of possessing a maximal independent set intersecting every non-augmentable path (in particular, every longest path). The classes are the arc-local tournament, quasi-transitive, locally in-semicomplete (out-semicomplete), and semicomplete k-partite digraphs. We present results on strongly internally and finally non-augmentable paths as well as a result that relates the degree of vertices and the length of longest paths. A short survey is included in the introduction
k-colored kernels
We study -colored kernels in -colored digraphs. An -colored digraph
has -colored kernel if there exists a subset of its vertices such
that
(i) from every vertex there exists an at most -colored
directed path from to a vertex of and
(ii) for every there does not exist an at most -colored
directed path between them.
In this paper, we prove that for every integer there exists a -colored digraph without -colored kernel and if every directed
cycle of an -colored digraph is monochromatic, then it has a -colored
kernel for every positive integer We obtain the following results for some
generalizations of tournaments:
(i) -colored quasi-transitive and 3-quasi-transitive digraphs have a %
-colored kernel for every and respectively (we conjecture
that every -colored -quasi-transitive digraph has a % -colored kernel
for every , and
(ii) -colored locally in-tournament (out-tournament, respectively)
digraphs have a -colored kernel provided that every arc belongs to a
directed cycle and every directed cycle is at most -colored
Tournaments with kernels by monochromatic paths
In this paper we prove the existence of kernels by monochromatic paths in m-coloured tournaments in which every cyclic tournament of order 3 is atmost 2-coloured in addition to other restrictions on the colouring ofcertain subdigraphs. We point out that in all previous results on kernelsby monochromatic paths in arc coloured tournaments, certain smallsubstructures are required to be monochromatic or monochromatic with atmost one exception, whereas here we allow up to three colours in two smallsubstructures
On the existence and number of -kings in -quasi-transitive digraphs
Let be a digraph and an integer. We say that
is -quasi-transitive if for every directed path in
, then or . Clearly, a
2-quasi-transitive digraph is a quasi-transitive digraph in the usual sense.
Bang-Jensen and Gutin proved that a quasi-transitive digraph has a 3-king
if and only if has a unique initial strong component and, if has a
3-king and the unique initial strong component of has at least three
vertices, then has at least three 3-kings. In this paper we prove the
following generalization: A -quasi-transitive digraph has a -king
if and only if has a unique initial strong component, and if has a
-king then, either all the vertices of the unique initial strong
components are -kings or the number of -kings in is at least
.Comment: 17 page
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