950 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
On retracts, absolute retracts, and folds in cographs
Let G and H be two cographs. We show that the problem to determine whether H
is a retract of G is NP-complete. We show that this problem is fixed-parameter
tractable when parameterized by the size of H. When restricted to the class of
threshold graphs or to the class of trivially perfect graphs, the problem
becomes tractable in polynomial time. The problem is also soluble when one
cograph is given as an induced subgraph of the other. We characterize absolute
retracts of cographs.Comment: 15 page
b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs
A b-coloring of a graph is a proper coloring such that every color class
contains a vertex that is adjacent to all other color classes. The b-chromatic
number of a graph G, denoted by \chi_b(G), is the maximum number t such that G
admits a b-coloring with t colors. A graph G is called b-continuous if it
admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and
b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of
G, and every induced subgraph H_2 of H_1.
We investigate the b-chromatic number of graphs with stability number two.
These are exactly the complements of triangle-free graphs, thus including all
complements of bipartite graphs. The main results of this work are the
following:
- We characterize the b-colorings of a graph with stability number two in
terms of matchings with no augmenting paths of length one or three. We derive
that graphs with stability number two are b-continuous and b-monotonic.
- We prove that it is NP-complete to decide whether the b-chromatic number of
co-bipartite graph is at most a given threshold.
- We describe a polynomial time dynamic programming algorithm to compute the
b-chromatic number of co-trees.
- Extending several previous results, we show that there is a polynomial time
dynamic programming algorithm for computing the b-chromatic number of
tree-cographs. Moreover, we show that tree-cographs are b-continuous and
b-monotonic
The 4-girth-thickness of the complete multipartite graph
The -girth-thickness of a graph is the smallest number
of planar subgraphs of girth at least whose union is . In this paper, we
calculate the -girth-thickness of the complete -partite
graph when each part has an even number of vertices.Comment: 6 pages, 1 figur
Forbidden subgraphs and complete partitions
A graph is called an -graph if its vertex set can be partitioned into
parts of size at most with at least one edge between any two parts. Let
be the minimum for which there exists an -free -graph.
In this paper we build on the work of Axenovich and Martin, obtaining improved
bounds on this function when is a complete bipartite graph, even cycle, or
tree. Some of these bounds are best possible up to a constant factor and
confirm a conjecture of Axenovich and Martin in several cases. We also
generalize this extremal problem to uniform hypergraphs and prove some initial
results in that setting
Native NIR-emitting single colour centres in CVD diamond
Single-photon sources are a fundamental element for developing quantum
technologies, and sources based on colour centres in diamonds are among the
most promising candidates. The well-known NV centres are characterized by
several limitations, thus few other defects have recently been considered. In
the present work, we characterize in detail native efficient single colour
centres emitting in the near infra-red in both standard IIa single-crystal and
electronic-grade polycrystalline commercial CVD diamond samples. In the former
case, a high-temperature annealing process in vacuum is necessary to induce the
formation/activation of luminescent centres with good emission properties,
while in the latter case the annealing process has marginal beneficial effects
on the number and performances of native centres in commercially available
samples. Although displaying significant variability in several photo physical
properties (emission wavelength, emission rate instabilities, saturation
behaviours), these centres generally display appealing photophysical properties
for applications as single photon sources: short lifetimes, high emission rates
and strongly polarized light. The native centres are tentatively attributed to
impurities incorporated in the diamond crystal during the CVD growth of
high-quality type IIa samples, and offer promising perspectives in
diamond-based photonics.Comment: 27 pages, 10 figures. Submitted to "New Journal of Phsyics",
NJP-100003.R
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