1,908 research outputs found
Complete Acyclic Colorings
We study two parameters that arise from the dichromatic number and the
vertex-arboricity in the same way that the achromatic number comes from the
chromatic number. The adichromatic number of a digraph is the largest number of
colors its vertices can be colored with such that every color induces an
acyclic subdigraph but merging any two colors yields a monochromatic directed
cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest
number of colors that can be used such that every color induces a forest but
merging any two yields a monochromatic cycle. We study the relation between
these parameters and their behavior with respect to other classical parameters
such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure
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
Arc-Disjoint Paths and Trees in 2-Regular Digraphs
An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected
spanning subdigraph of D in which every vertex x != s has precisely one arc
entering (leaving) it and s has no arcs entering (leaving) it. We settle the
complexity of the following two problems:
1) Given a 2-regular digraph , decide if it contains two arc-disjoint
branchings B^+_u, B^-_v.
2) Given a 2-regular digraph D, decide if it contains an out-branching B^+_u
such that D remains connected after removing the arcs of B^+_u.
Both problems are NP-complete for general digraphs. We prove that the first
problem remains NP-complete for 2-regular digraphs, whereas the second problem
turns out to be polynomial when we do not prescribe the root in advance. We
also prove that, for 2-regular digraphs, the latter problem is in fact
equivalent to deciding if contains two arc-disjoint out-branchings. We
generalize this result to k-regular digraphs where we want to find a number of
pairwise arc-disjoint spanning trees and out-branchings such that there are k
in total, again without prescribing any roots.Comment: 9 pages, 7 figure
On a -ultrahomogeneous oriented graph
The notion of a -ultrahomogeneous graph, due to Isaksen et al.,
is adapted for digraphs, and subsequently a strongly connected
-ultrahomogeneous oriented graph on 168 vertices and 126 pairwise
arc-disjoint 4-cycles is presented, with regular indegree and outdegree 3 and
no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph
via pencils of ordered lines of the Fano plane in which pencils are replaced by
ordered pencils.Comment: 4 pages, 2 figures, 2 table
Generalized Interlinked Cycle Cover for Index Coding
A source coding problem over a noiseless broadcast channel where the source
is pre-informed about the contents of the cache of all receivers, is an index
coding problem. Furthermore, if each message is requested by one receiver, then
we call this an index coding problem with a unicast message setting. This
problem can be represented by a directed graph. In this paper, we first define
a structure (we call generalized interlinked cycles (GIC)) in directed graphs.
A GIC consists of cycles which are interlinked in some manner (i.e., not
disjoint), and it turns out that the GIC is a generalization of cliques and
cycles. We then propose a simple scalar linear encoding scheme with linear time
encoding complexity. This scheme exploits GICs in the digraph. We prove that
our scheme is optimal for a class of digraphs with message packets of any
length. Moreover, we show that our scheme can outperform existing techniques,
e.g., partial clique cover, local chromatic number, composite-coding, and
interlinked cycle cover.Comment: Extended version of the paper which is to be presented at the IEEE
Information Theory Workshop (ITW), 2015 Jej
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