72 research outputs found
On homomorphism of oriented graphs with respect to push operation
An oriented graph is a directed graph without any cycle of length at most 2.
To push a vertex of a directed graph is to reverse the orientation of the arcs
incident to that vertex. Klostermeyer and MacGillivray defined push graphs
which are equivalence class of oriented graphs with respect to vertex pushing
operation. They studied the homomorphism of the equivalence classes of oriented
graphs with respect to push operation. In this article, we further study the
same topic and answer some of the questions asked in the above mentioned work.
The anti-twinned graph of an oriented graph is obtained by adding and pushing a
copy of each of its vertices. In particular, we show that two oriented graphs
are in a push relation if and only if they have isomorphic anti-twinned graphs.
Moreover, we study oriented homomorphisms of outerplanar graphs with girth at
least five, planar graphs and planar graphs with girth at least eight with
respect to the push operation
Erratum for "On oriented cliques with respect to push operation"
An error is spotted in the statement of Theorem~1.3 of our published article
titled "On oriented cliques with respect to push operation" (Discrete Applied
Mathematics 2017). The theorem provided an exhaustive list of 16 minimal (up to
spanning subgraph inclusion) underlying planar push cliques. The error was
that, one of the 16 graphs from the above list was missing an arc. We correct
the error and restate the corrected statement in this article. We also point
out the reason for the error and comment that the error occurred due to a
mistake in a particular lemma. We present the corrected proof of that
particular lemma as well. Moreover, a few counts were wrongly reported due to
the above mentioned error. So we update our reported counts after correction in
this article
On chromatic number of colored mixed graphs
An -colored mixed graph is a graph with its arcs having one of the
different colors and edges having one of the different colors. A
homomorphism of an -colored mixed graph to an -colored
mixed graph is a vertex mapping such that if is an arc (edge) of color
in , then is an arc (edge) of color in . The
\textit{-colored mixed chromatic number} of an
-colored mixed graph is the order (number of vertices) of the
smallest homomorphic image of . This notion was introduced by
Ne\v{s}et\v{r}il and Raspaud (2000, J. Combin. Theory, Ser. B 80, 147--155).
They showed that where is a
-acyclic colorable graph. We proved the tightness of this bound. We also
showed that the acyclic chromatic number of a graph is bounded by if its -colored mixed
chromatic number is at most .
Furthermore, using probabilistic method, we showed that for graphs with
maximum degree its -colored mixed chromatic number is at most
. In particular, the last result directly
improves the upper bound of oriented chromatic number of
graphs with maximum degree , obtained by Kostochka, Sopena and Zhu
(1997, J. Graph Theory 24, 331--340) to . We also
show that there exists a graph with maximum degree and -colored
mixed chromatic number at least
Outerplanar and planar oriented cliques
The clique number of an undirected graph is the maximum order of a
complete subgraph of and is a well-known lower bound for the chromatic
number of . Every proper -coloring of may be viewed as a homomorphism
(an edge-preserving vertex mapping) of to the complete graph of order .
By considering homomorphisms of oriented graphs (digraphs without cycles of
length at most 2), we get a natural notion of (oriented) colorings and oriented
chromatic number of oriented graphs. An oriented clique is then an oriented
graph whose number of vertices and oriented chromatic number coincide. However,
the structure of oriented cliques is much less understood than in the
undirected case.
In this paper, we study the structure of outerplanar and planar oriented
cliques. We first provide a list of 11 graphs and prove that an outerplanar
graph can be oriented as an oriented clique if and only if it contains one of
these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured
that the order of a planar oriented clique is at most 15, which was later
proved by Sen [S. Sen. Maximum Order of a Planar Oclique Is 15. Proc.
IWOCA'2012. {\em Lecture Notes Comput. Sci.} 7643:130--142]. We show that any
planar oriented clique on 15 vertices must contain a particular oriented graph
as a spanning subgraph, thus reproving the above conjecture. We also provide
tight upper bounds for the order of planar oriented cliques of girth for
all
On oriented cliques with respect to push operation
To push a vertex of a directed graph is to change
the orientations of all the arcs incident with . An oriented graph is a
directed graph without any cycle of length at most 2. An oriented clique is an
oriented graph whose non-adjacent vertices are connected by a directed 2-path.
A push clique is an oriented clique that remains an oriented clique even if one
pushes any set of vertices of it. We show that it is NP-complete to decide if
an undirected graph is underlying graph of a push clique or not. We also prove
that a planar push clique can have at most 8 vertices. We also provide an
exhaustive list of minimal (with respect to spanning subgraph inclusion) planar
push cliques
Homomorphisms of signed planar graphs
Signed graphs are studied since the middle of the last century. Recently, the
notion of homomorphism of signed graphs has been introduced since this notion
captures a number of well known conjectures which can be reformulated using the
definitions of signed homomorphism.
In this paper, we introduce and study the properties of some target graphs
for signed homomorphism. Using these properties, we obtain upper bounds on the
signed chromatic numbers of graphs with bounded acyclic chromatic number and of
signed planar graphs with given girth
Walk-powers and homomorphism bound of planar graphs
As an extension of the Four-Color Theorem it is conjectured that every planar
graph of odd-girth at least admits a homomorphism to
where 's are
standard basis and is all 1 vector. Noting that itself is of
odd-girth , in this work we show that if the conjecture is true, then
is an optimal such a graph both with respect to number of vertices
and number of edges. The result is obtained using the notion of walk-power of
graphs and their clique numbers.
An analogous result is proved for bipartite signed planar graphs of
unbalanced-girth . The work is presented on a uniform frame work of planar
consistent signed graphs
Analogous to cliques for (m,n)-colored mixed graphs
Vertex coloring of a graph with -colors can be equivalently thought to
be a graph homomorphism (edge preserving vertex mapping) of to the complete
graph of order . So, in that sense, the chromatic number of
will be the order of the smallest complete graph to which admits a
homomorphism to. As every graph, which is not a complete graph, admits a
homomorphism to a smaller complete graph, we can redefine the chromatic number
of to be the order of the smallest graph to which admits a
homomorphism to. Of course, such a smallest graph must be a complete graph as
they are the only graphs with chromatic number equal to their order.
The concept of vertex coloring can be generalize for other types of graphs.
Naturally, the chromatic number is defined to be the order of the smallest
graph (of the same type) to which a graph admits homomorphism to. The analogous
notion of clique turns out to be the graphs with order equal to their (so
defined) "chromatic number". These "cliques" turns out to be much more
complicated than their undirected counterpart and are interesting objects of
study. In this article, we mainly study different aspects of "cliques" for
signed (graphs with positive or negative signs assigned to each edge) and
switchable signed graphs (equivalence class of signed graph with respect to
switching signs of edges incident to the same vertex).Comment: arXiv admin note: substantial text overlap with arXiv:1411.719
On relative clique number of colored mixed graphs
An -colored mixed graph is a graph having arcs of different
colors and edges of different colors. A graph homomorphism of an )-colored mixed graph to an -colored mixed graph is a vertex
mapping such that if is an arc (edge) of color in , then
is also an arc (edge) of color . The (-colored mixed chromatic number
of an -colored mixed graph , introduced by Ne\v{s}et\v{r}il and
Raspaud [J. Combin. Theory Ser. B 2000] is the order (number of vertices) of
the smallest homomorphic image of . Later Bensmail, Duffy and Sen [Graphs
Combin. 2017] introduced another parameter related to the -colored
mixed chromatic number, namely, the -relative clique number as the
maximum cardinality of a vertex subset which, pairwise, must have distinct
images with respect to any colored homomorphism.
In this article, we study the )-relative clique number for the family
of subcubic graphs, graphs with maximum degree , planar graphs and
triangle-free planar graphs and provide new improved bounds in each of the
cases. In particular, for subcubic graphs we provide exact value of the
parameter
Chromatic number of signed graphs with bounded maximum degree
A signed graph is a graph positive and negative (
denotes the set of negative edges). To re-sign a vertex of a signed graph is to switch the signs of the edges incident to . If one can
obtain by re-signing some vertices of , then
.
A signed graphs admits an homomorphism to if
there is a sign preserving vertex mapping from to
for some .
The signed chromatic number of the signed graph is the minimum order (number of vertices) of a signed graph such that admits a homomorphism to . For
a family of signed graphs .
We prove for all where is the
family of connected signed graphs with maximum degree . \end{abstract
- …