112 research outputs found
Colourings of cubic graphs inducing isomorphic monochromatic subgraphs
A -bisection of a bridgeless cubic graph is a -colouring of its
vertex set such that the colour classes have the same cardinality and all
connected components in the two subgraphs induced by the colour classes
(monochromatic components in what follows) have order at most . Ban and
Linial conjectured that every bridgeless cubic graph admits a -bisection
except for the Petersen graph. A similar problem for the edge set of cubic
graphs has been studied: Wormald conjectured that every cubic graph with
has a -edge colouring such that the two
monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose
components are paths). Finally, Ando conjectured that every cubic graph admits
a bisection such that the two induced monochromatic subgraphs are isomorphic.
In this paper, we give a detailed insight into the conjectures of Ban-Linial
and Wormald and provide evidence of a strong relation of both of them with
Ando's conjecture. Furthermore, we also give computational and theoretical
evidence in their support. As a result, we pose some open problems stronger
than the above mentioned conjectures. Moreover, we prove Ban-Linial's
conjecture for cubic cycle permutation graphs.
As a by-product of studying -edge colourings of cubic graphs having linear
forests as monochromatic components, we also give a negative answer to a
problem posed by Jackson and Wormald about certain decompositions of cubic
graphs into linear forests.Comment: 33 pages; submitted for publicatio
On the Complexity of Digraph Colourings and Vertex Arboricity
It has been shown by Bokal et al. that deciding 2-colourability of digraphs
is an NP-complete problem. This result was later on extended by Feder et al. to
prove that deciding whether a digraph has a circular -colouring is
NP-complete for all rational . In this paper, we consider the complexity
of corresponding decision problems for related notions of fractional colourings
for digraphs and graphs, including the star dichromatic number, the fractional
dichromatic number and the circular vertex arboricity. We prove the following
results:
Deciding if the star dichromatic number of a digraph is at most is
NP-complete for every rational .
Deciding if the fractional dichromatic number of a digraph is at most is
NP-complete for every .
Deciding if the circular vertex arboricity of a graph is at most is
NP-complete for every rational .
To show these results, different techniques are required in each case. In
order to prove the first result, we relate the star dichromatic number to a new
notion of homomorphisms between digraphs, called circular homomorphisms, which
might be of independent interest. We provide a classification of the
computational complexities of the corresponding homomorphism colouring problems
similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur
Compact Labelings For Efficient First-Order Model-Checking
We consider graph properties that can be checked from labels, i.e., bit
sequences, of logarithmic length attached to vertices. We prove that there
exists such a labeling for checking a first-order formula with free set
variables in the graphs of every class that is \emph{nicely locally
cwd-decomposable}. This notion generalizes that of a \emph{nicely locally
tree-decomposable} class. The graphs of such classes can be covered by graphs
of bounded \emph{clique-width} with limited overlaps. We also consider such
labelings for \emph{bounded} first-order formulas on graph classes of
\emph{bounded expansion}. Some of these results are extended to counting
queries
Decomposing cubic graphs into isomorphic linear forests
A common problem in graph colouring seeks to decompose the edge set of a
given graph into few similar and simple subgraphs, under certain divisibility
conditions. In 1987 Wormald conjectured that the edges of every cubic graph on
vertices can be partitioned into two isomorphic linear forests. We prove
this conjecture for large connected cubic graphs. Our proof uses a wide range
of probabilistic tools in conjunction with intricate structural analysis, and
introduces a variety of local recolouring techniques.Comment: 49 pages, many figure
Inapproximability of maximal strip recovery
In comparative genomic, the first step of sequence analysis is usually to
decompose two or more genomes into syntenic blocks that are segments of
homologous chromosomes. For the reliable recovery of syntenic blocks, noise and
ambiguities in the genomic maps need to be removed first. Maximal Strip
Recovery (MSR) is an optimization problem proposed by Zheng, Zhu, and Sankoff
for reliably recovering syntenic blocks from genomic maps in the midst of noise
and ambiguities. Given genomic maps as sequences of gene markers, the
objective of \msr{d} is to find subsequences, one subsequence of each
genomic map, such that the total length of syntenic blocks in these
subsequences is maximized. For any constant , a polynomial-time
2d-approximation for \msr{d} was previously known. In this paper, we show that
for any , \msr{d} is APX-hard, even for the most basic version of the
problem in which all gene markers are distinct and appear in positive
orientation in each genomic map. Moreover, we provide the first explicit lower
bounds on approximating \msr{d} for all . In particular, we show that
\msr{d} is NP-hard to approximate within . From the other
direction, we show that the previous 2d-approximation for \msr{d} can be
optimized into a polynomial-time algorithm even if is not a constant but is
part of the input. We then extend our inapproximability results to several
related problems including \cmsr{d}, \gapmsr{\delta}{d}, and
\gapcmsr{\delta}{d}.Comment: A preliminary version of this paper appeared in two parts in the
Proceedings of the 20th International Symposium on Algorithms and Computation
(ISAAC 2009) and the Proceedings of the 4th International Frontiers of
Algorithmics Workshop (FAW 2010
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