603 research outputs found

### Quantum walks on general graphs

Quantum walks, both discrete (coined) and continuous time, on a general graph
of N vertices with undirected edges are reviewed in some detail. The resource
requirements for implementing a quantum walk as a program on a quantum computer
are compared and found to be very similar for both discrete and continuous time
walks. The role of the oracle, and how it changes if more prior information
about the graph is available, is also discussed.Comment: 8 pages, v2: substantial rewrite improves clarity, corrects errors
and omissions; v3: removes major error in final section and integrates
remainder into other sections, figures remove

### Maximum $\Delta$-edge-colorable subgraphs of class II graphs

A graph $G$ is class II, if its chromatic index is at least $\Delta+1$. Let
$H$ be a maximum $\Delta$-edge-colorable subgraph of $G$. The paper proves best
possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of
maximum $\Delta$-edge-colorable subgraphs. It is shown that every set of
vertex-disjoint cycles of a class II graph with $\Delta\geq3$ can be extended
to a maximum $\Delta$-edge-colorable subgraph. Simple graphs have a maximum
$\Delta$-edge-colorable subgraph such that the complement is a matching.
Furthermore, a maximum $\Delta$-edge-colorable subgraph of a simple graph is
always class I.Comment: 13 pages, 2 figures, the proof of the Lemma 1 is correcte

### The chromatic index of graphs with large maximum degree

By Vizing's Theorem, any graph G has chromatic index equal either to its maximum degree [Delta](G) or [Delta](G) + 1. A simple method is given for determining exactly the chromatic index of any graph with 2s + 2 vertices and maximum degree 2s.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25365/1/0000813.pd

### Representing Partitions on Trees

In evolutionary biology, biologists often face the problem of constructing a phylogenetic tree on a set X of species from a multiset Î of partitions corresponding to various attributes of these species. One approach that is used to solve this problem is to try instead to associate a tree (or even a network) to the multiset Î£Î consisting of all those bipartitions {A,X âˆ’ A} with A a part of some partition in Î . The rational behind this approach is that a phylogenetic tree with leaf set X can be uniquely represented by the set of bipartitions of X induced by its edges. Motivated by these considerations, given a multiset Î£ of bipartitions corresponding to a phylogenetic tree on X, in this paper we introduce and study the set P(Î£) consisting of those multisets of partitions Î of X with Î£Î = Î£. More specifically, we characterize when P(Î£) is non-empty, and also identify some partitions in P(Î£) that are of maximum and minimum size. We also show that it is NP-complete to decide when P(Î£) is non-empty in case Î£ is an arbitrary multiset of bipartitions of X. Ultimately, we hope that by gaining a better understanding of the mapping that takes an arbitrary partition system Î to the multiset Î£Î , we will obtain new insights into the use of median networks and, more generally, split-networks to visualize sets of partitions

### Computation by measurements: a unifying picture

The ability to perform a universal set of quantum operations based solely on
static resources and measurements presents us with a strikingly novel viewpoint
for thinking about quantum computation and its powers. We consider the two
major models for doing quantum computation by measurements that have hitherto
appeared in the literature and show that they are conceptually closely related
by demonstrating a systematic local mapping between them. This way we
effectively unify the two models, showing that they make use of interchangeable
primitives. With the tools developed for this mapping, we then construct more
resource-effective methods for performing computation within both models and
propose schemes for the construction of arbitrary graph states employing
two-qubit measurements alone.Comment: 13 pages, 18 figures, REVTeX

### List precoloring extension in planar graphs

A celebrated result of Thomassen states that not only can every planar graph
be colored properly with five colors, but no matter how arbitrary palettes of
five colors are assigned to vertices, one can choose a color from the
corresponding palette for each vertex so that the resulting coloring is proper.
This result is referred to as 5-choosability of planar graphs. Albertson asked
whether Thomassen's theorem can be extended by precoloring some vertices which
are at a large enough distance apart in a graph. Here, among others, we answer
the question in the case when the graph does not contain short cycles
separating precolored vertices and when there is a "wide" Steiner tree
containing all the precolored vertices.Comment: v2: 15 pages, 11 figres, corrected typos and new proof of Theorem
3(2

### Approximating the Minimum Equivalent Digraph

The MEG (minimum equivalent graph) problem is, given a directed graph, to
find a small subset of the edges that maintains all reachability relations
between nodes. The problem is NP-hard. This paper gives an approximation
algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its
analysis are based on the simple idea of contracting long cycles. (This result
is strengthened slightly in ``On strongly connected digraphs with bounded cycle
length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local
improvement'' algorithm, showing that its performance guarantee is 1.75.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms
(1994

### Fractional total colourings of graphs of high girth

Reed conjectured that for every epsilon>0 and Delta there exists g such that
the fractional total chromatic number of a graph with maximum degree Delta and
girth at least g is at most Delta+1+epsilon. We prove the conjecture for
Delta=3 and for even Delta>=4 in the following stronger form: For each of these
values of Delta, there exists g such that the fractional total chromatic number
of any graph with maximum degree Delta and girth at least g is equal to
Delta+1

### A new approach to nonrepetitive sequences

A sequence is nonrepetitive if it does not contain two adjacent identical
blocks. The remarkable construction of Thue asserts that 3 symbols are enough
to build an arbitrarily long nonrepetitive sequence. It is still not settled
whether the following extension holds: for every sequence of 3-element sets
$L_1,..., L_n$ there exists a nonrepetitive sequence $s_1, ..., s_n$ with
$s_i\in L_i$. Applying the probabilistic method one can prove that this is true
for sufficiently large sets $L_i$. We present an elementary proof that sets of
size 4 suffice (confirming the best known bound). The argument is a simple
counting with Catalan numbers involved. Our approach is inspired by a new
algorithmic proof of the Lov\'{a}sz Local Lemma due to Moser and Tardos and its
interpretations by Fortnow and Tao. The presented method has further
applications to nonrepetitive games and nonrepetitive colorings of graphs.Comment: 5 pages, no figures.arXiv admin note: substantial text overlap with
arXiv:1103.381

### 5-list-coloring planar graphs with distant precolored vertices

We answer positively the question of Albertson asking whether every planar
graph can be $5$-list-colored even if it contains precolored vertices, as long
as they are sufficiently far apart from each other. In order to prove this
claim, we also give bounds on the sizes of graphs critical with respect to
5-list coloring. In particular, if G is a planar graph, H is a connected
subgraph of G and L is an assignment of lists of colors to the vertices of G
such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G
contains a subgraph with O(|H|^2) vertices that is not L-colorable.Comment: 53 pages, 9 figures version 2: addresses suggestions by reviewer

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