7,396 research outputs found
Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes
A hole in a graph is an induced subgraph which is a cycle of length at least
four. We prove that for every positive integer k, every triangle-free graph
with sufficiently large chromatic number contains holes of k consecutive
lengths
The Computational Complexity of Linear Optics
We give new evidence that quantum computers -- moreover, rudimentary quantum
computers built entirely out of linear-optical elements -- cannot be
efficiently simulated by classical computers. In particular, we define a model
of computation in which identical photons are generated, sent through a
linear-optical network, then nonadaptively measured to count the number of
photons in each mode. This model is not known or believed to be universal for
quantum computation, and indeed, we discuss the prospects for realizing the
model using current technology. On the other hand, we prove that the model is
able to solve sampling problems and search problems that are classically
intractable under plausible assumptions. Our first result says that, if there
exists a polynomial-time classical algorithm that samples from the same
probability distribution as a linear-optical network, then P^#P=BPP^NP, and
hence the polynomial hierarchy collapses to the third level. Unfortunately,
this result assumes an extremely accurate simulation. Our main result suggests
that even an approximate or noisy classical simulation would already imply a
collapse of the polynomial hierarchy. For this, we need two unproven
conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is
#P-hard to approximate the permanent of a matrix A of independent N(0,1)
Gaussian entries, with high probability over A; and the "Permanent
Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with
high probability over A. We present evidence for these conjectures, both of
which seem interesting even apart from our application. This paper does not
assume knowledge of quantum optics. Indeed, part of its goal is to develop the
beautiful theory of noninteracting bosons underlying our model, and its
connection to the permanent function, in a self-contained way accessible to
theoretical computer scientists.Comment: 94 pages, 4 figure
Induced subgraphs of graphs with large chromatic number. XIII. New brooms
Gy\'arf\'as and Sumner independently conjectured that for every tree , the
class of graphs not containing as an induced subgraph is -bounded,
that is, the chromatic numbers of graphs in this class are bounded above by a
function of their clique numbers. This remains open for general trees , but
has been proved for some particular trees. For , let us say a broom of
length is a tree obtained from a -edge path with ends by adding
some number of leaves adjacent to , and we call its handle. A tree
obtained from brooms of lengths by identifying their handles is a
-multibroom. Kierstead and Penrice proved that every
-multibroom satisfies the Gy\'arf\'as-Sumner conjecture, and
Kierstead and Zhu proved the same for -multibrooms. In this paper
give a common generalization: we prove that every
-multibroom satisfies the Gy\'arf\'as-Sumner conjecture
Random graphs from a block-stable class
A class of graphs is called block-stable when a graph is in the class if and
only if each of its blocks is. We show that, as for trees, for most -vertex
graphs in such a class, each vertex is in at most blocks, and each path passes through at most blocks.
These results extend to `weakly block-stable' classes of graphs
On lower bounds for the matching number of subcubic graphs
We give a complete description of the set of triples (a,b,c) of real numbers
with the following property. There exists a constant K such that a n_3 + b n_2
+ c n_1 - K is a lower bound for the matching number of every connected
subcubic graph G, where n_i denotes the number of vertices of degree i for each
i
Maximising the number of induced cycles in a graph
We determine the maximum number of induced cycles that can be contained in a
graph on vertices, and show that there is a unique graph that
achieves this maximum. This answers a question of Tuza. We also determine the
maximum number of odd or even cycles that can be contained in a graph on vertices and characterise the extremal graphs. This resolves a conjecture
of Chv\'atal and Tuza from 1988.Comment: 36 page
Maximising -Colourings of Graphs
For graphs and , an -colouring of is a map
such that . The number of -colourings of is denoted by .
We prove the following: for all graphs and , there is a
constant such that, if , the graph
maximises the number of -colourings among all
connected graphs with vertices and minimum degree . This answers a
question of Engbers.
We also disprove a conjecture of Engbers on the graph that maximises the
number of -colourings when the assumption of the connectivity of is
dropped.
Finally, let be a graph with maximum degree . We show that, if
does not contain the complete looped graph on vertices or as a
component and , then the following holds: for
sufficiently large, the graph maximises the number of
-colourings among all graphs on vertices with minimum degree .
This partially answers another question of Engbers
Intersections of hypergraphs
Given two weighted k-uniform hypergraphs G, H of order n, how much (or
little) can we make them overlap by placing them on the same vertex set? If we
place them at random, how concentrated is the distribution of the intersection?
The aim of this paper is to investigate these questions
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