285 research outputs found
Oriented coloring on recursively defined digraphs
Coloring is one of the most famous problems in graph theory. The coloring
problem on undirected graphs has been well studied, whereas there are very few
results for coloring problems on directed graphs. An oriented k-coloring of an
oriented graph G=(V,A) is a partition of the vertex set V into k independent
sets such that all the arcs linking two of these subsets have the same
direction. The oriented chromatic number of an oriented graph G is the smallest
k such that G allows an oriented k-coloring. Deciding whether an acyclic
digraph allows an oriented 4-coloring is NP-hard. It follows, that finding the
chromatic number of an oriented graph is an NP-hard problem. This motivates to
consider the problem on oriented co-graphs. After giving several
characterizations for this graph class, we show a linear time algorithm which
computes an optimal oriented coloring for an oriented co-graph. We further
prove how the oriented chromatic number can be computed for the disjoint union
and order composition from the oriented chromatic number of the involved
oriented co-graphs. It turns out that within oriented co-graphs the oriented
chromatic number is equal to the length of a longest oriented path plus one. We
also show that the graph isomorphism problem on oriented co-graphs can be
solved in linear time.Comment: 14 page
Rainbow eulerian multidigraphs and the product of cycles
An arc colored eulerian multidigraph with colors is rainbow eulerian if
there is an eulerian circuit in which a sequence of colors repeats. The
digraph product that refers the title was introduced by Figueroa-Centeno et al.
as follows: let be a digraph and let be a family of digraphs such
that for every . Consider any function
. Then the product is the
digraph with vertex set and if and only if and .
In this paper we use rainbow eulerian multidigraphs and permutations as a way
to characterize the -product of oriented cycles. We study the
behavior of the -product when applied to digraphs with unicyclic
components. The results obtained allow us to get edge-magic labelings of graphs
formed by the union of unicyclic components and with different magic sums.Comment: 12 pages, 5 figure
Acyclic Subgraphs of Planar Digraphs
An acyclic set in a digraph is a set of vertices that induces an acyclic
subgraph. In 2011, Harutyunyan conjectured that every planar digraph on
vertices without directed 2-cycles possesses an acyclic set of size at least
. We prove this conjecture for digraphs where every directed cycle has
length at least 8. More generally, if is the length of the shortest
directed cycle, we show that there exists an acyclic set of size at least .Comment: 9 page
Rainbow eulerian multidigraphs and the product of cycles
An arc colored eulerian multidigraph with colors is rainbow eulerian if there is an eulerian circuit in which a sequence of colors repeats. The digraph product that refers the title was introduced by Figueroa-Centeno et al. as follows: let be a digraph and let be a family of digraphs such that for every . Consider any function . Then the product is the digraph with vertex set and if and only if and . In this paper we use rainbow eulerian multidigraphs and permutations as a way to characterize the -product of oriented cycles. We study the behavior of the -product when applied to digraphs with unicyclic components. The results obtained allow us to get edge-magic labelings of graphs formed by the union of unicyclic components and with different magic sums.Supported by the Spanish Research Council under project MTM2011-28800-C02-01 and by the Catalan Research Council under grant 2009SGR1387
A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension
We study the extension of the Kechris-Solecki-Todorcevic dichotomy on
analytic graphs to dimensions higher than 2. We prove that the extension is
possible in any dimension, finite or infinite. The original proof works in the
case of the finite dimension. We first prove that the natural extension does
not work in the case of the infinite dimension, for the notion of continuous
homomorphism used in the original theorem. Then we solve the problem in the
case of the infinite dimension. Finally, we prove that the natural extension
works in the case of the infinite dimension, but for the notion of
Baire-measurable homomorphism
- …