174 research outputs found
Colouring exact distance graphs of chordal graphs
For a graph and positive integer , the exact distance- graph
is the graph with vertex set and with an edge between
vertices and if and only if and have distance . Recently,
there has been an effort to obtain bounds on the chromatic number
of exact distance- graphs for from certain
classes of graphs. In particular, if a graph has tree-width , it has
been shown that for odd ,
and for even . We
show that if is chordal and has tree-width , then for odd , and for even .
If we could show that for every graph of tree-width there is a
chordal graph of tree-width which contains as an isometric subgraph
(i.e., a distance preserving subgraph), then our results would extend to all
graphs of tree-width . While we cannot do this, we show that for every graph
of genus there is a graph which is a triangulation of genus and
contains as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which
arise from reviewers' comment
Clique immersions in graphs of independence number two with certain forbidden subgraphs
The Lescure-Meyniel conjecture is the analogue of Hadwiger's conjecture for
the immersion order. It states that every graph contains the complete graph
as an immersion, and like its minor-order counterpart it is open
even for graphs with independence number 2. We show that every graph with
independence number and no hole of length between and
satisfies this conjecture. In particular, every -free graph
with satisfies the Lescure-Meyniel conjecture. We give
another generalisation of this corollary, as follows. Let and be graphs
with independence number at most 2, such that . If is
-free, then satisfies the Lescure-Meyniel conjecture.Comment: 14 pages, 3 figures. The statements of lemmas 3.1, 4.1, and 4.2 are
slightly changed from the previous version in order to fix some minor errors
in the proofs of theorems 3.2 and 4.3. Shorter proof of Proposition 5.2 give
Quadratic Dynamical Decoupling with Non-Uniform Error Suppression
We analyze numerically the performance of the near-optimal quadratic
dynamical decoupling (QDD) single-qubit decoherence errors suppression method
[J. West et al., Phys. Rev. Lett. 104, 130501 (2010)]. The QDD sequence is
formed by nesting two optimal Uhrig dynamical decoupling sequences for two
orthogonal axes, comprising N1 and N2 pulses, respectively. Varying these
numbers, we study the decoherence suppression properties of QDD directly by
isolating the errors associated with each system basis operator present in the
system-bath interaction Hamiltonian. Each individual error scales with the
lowest order of the Dyson series, therefore immediately yielding the order of
decoherence suppression. We show that the error suppression properties of QDD
are dependent upon the parities of N1 and N2, and near-optimal performance is
achieved for general single-qubit interactions when N1=N2.Comment: 17 pages, 22 figure
Chromatic and structural properties of sparse graph classes
A graph is a mathematical structure consisting of a set of objects, which we call vertices, and links between pairs of objects, which we call edges. Graphs are used to model many problems arising in areas such as physics, sociology, and computer science.
It is partially because of the simplicity of the definition of a graph that the concept can be so widely used. Nevertheless, when applied to a particular task, it is not always necessary to study graphs in all their generality, and it can be convenient to studying them from a restricted point of view. Restriction can come from requiring graphs to be embeddable in a particular surface, to admit certain types of decompositions, or by forbidding some substructure. A collection of graphs satisfying a fixed restriction forms a class of graphs.
Many important classes of graphs satisfy that graphs belonging to it cannot have many edges in comparison with the number of vertices. Such is the case of classes with an upper bound on the maximum degree, and of classes excluding a fixed minor. Recently, the notion of classes with bounded expansion was introduced by Neˇsetˇril and Ossona de Mendez [62], as a generalisation of many important types of sparse classes. In this thesis we study chromatic and structural properties of classes with bounded expansion.
We say a graph is k-degenerate if each of its subgraphs has a vertex of degree at most k. The degeneracy is thus a measure of the density of a graph. This notion has been generalised with the introduction, by Kierstead and Yang [47], of the generalised colouring numbers. These parameters have found applications in many areas of Graph Theory, including a characterisation of classes with bounded expansion. One of the main results of this thesis is a series of upper bounds on the generalised colouring numbers, for different sparse classes of graphs, such as classes excluding a fixed complete minor, classes with bounded genus and classes with bounded tree-width.
We also study the following problem: for a fixed positive integer p, how many colours do we need to colour a given graph in such a way that vertices at distance exactly p get different colours? When considering classes with bounded expansion, we improve dramatically on the previously known upper bounds for the number of colours needed.
Finally, we introduce a notion of addition of graph classes, and show various cases in which sparse classes can be summed so as to obtain another sparse class
High Fidelity Adiabatic Quantum Computation via Dynamical Decoupling
We introduce high-order dynamical decoupling strategies for open system
adiabatic quantum computation. Our numerical results demonstrate that a
judicious choice of high-order dynamical decoupling method, in conjunction with
an encoding which allows computation to proceed alongside decoupling, can
dramatically enhance the fidelity of adiabatic quantum computation in spite of
decoherence.Comment: 5 pages, 4 figure
Universal arrays
A word on symbols is a sequence of letters from a fixed alphabet of size
. For an integer , we say that a word is -universal if, given
an arbitrary word of length , one can obtain it by removing entries from
. It is easily seen that the minimum length of a -universal word on
symbols is exactly . We prove that almost every word of size
is -universal with high probability, where is an explicit constant
whose value is roughly . Moreover, we show that the -universality
property for uniformly chosen words exhibits a sharp threshold. Finally, by
extending techniques of Alon [Geometric and Functional Analysis 27 (2017), no.
1, 1--32], we give asymptotically tight bounds for every higher dimensional
analogue of this problem.Comment: 12 page
Chromatic numbers of exact distance graphs
For any graph G = (V;E) and positive integer p, the exact distance-p graph G[\p] is the graph with vertex set V , which has an edge between vertices x and y if and only if x and y have distance p in G. For odd p, Nešetřil and Ossona de Mendez proved that for any fixed graph class with bounded expansion, the chromatic number of G[\p] is bounded by an absolute constant. Using the notion of generalised colouring numbers, we give a much simpler proof for the result of Nešetřil and Ossona de Mendez, which at the same time gives significantly better bounds. In particular, we show that for any graph G and odd positive integer p, the chromatic number of G[\p] is bounded by the weak (2
Characterizing and recognizing exact-distance squares of graphs
For a graph , its exact-distance square, , is the
graph with vertex set and with an edge between vertices and if and
only if and have distance (exactly) in . The graph is an
exact-distance square root of . We give a characterization of
graphs having an exact-distance square root, our characterization easily
leading to a polynomial-time recognition algorithm. We show that it is
NP-complete to recognize graphs with a bipartite exact-distance square root.
These two results strongly contrast known results on (usual) graph squares. We
then characterize graphs having a tree as an exact-distance square root, and
from this obtain a polynomial-time recognition algorithm for these graphs.
Finally, we show that, unlike for usual square roots, a graph might have
(arbitrarily many) non-isomorphic exact-distance square roots which are trees.Comment: 15 pages, 6 figure
Balanced-chromatic number and Hadwiger-like conjectures
Motivated by different characterizations of planar graphs and the 4-Color
Theorem, several structural results concerning graphs of high chromatic number
have been obtained. Toward strengthening some of these results, we consider the
\emph{balanced chromatic number}, , of a signed graph
. This is the minimum number of parts into which the vertices of a
signed graph can be partitioned so that none of the parts induces a negative
cycle. This extends the notion of the chromatic number of a graph since
, where denotes the signed graph
obtained from~ by replacing each edge with a pair of (parallel) positive and
negative edges. We introduce a signed version of Hadwiger's conjecture as
follows.
Conjecture: If a signed graph has no negative loop and no
-minor, then its balanced chromatic number is at most .
We prove that this conjecture is, in fact, equivalent to Hadwiger's
conjecture and show its relation to the Odd Hadwiger Conjecture.
Motivated by these results, we also consider the relation between
subdivisions and balanced chromatic number. We prove that if has
no negative loop and no -subdivision, then it admits a balanced
-coloring. This qualitatively generalizes a result of
Kawarabayashi (2013) on totally odd subdivisions
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