1,647 research outputs found
-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum
A -WORM coloring of a graph is an assignment of colors to the
vertices in such a way that the vertices of each -subgraph of get
precisely two colors. We study graphs which admit at least one such
coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014)
161--173] who asked whether every such graph has a -WORM coloring with two
colors. In fact for every integer there exists a -WORM colorable
graph in which the minimum number of colors is exactly . There also exist
-WORM colorable graphs which have a -WORM coloring with two colors
and also with colors but no coloring with any of colors. We
also prove that it is NP-hard to determine the minimum number of colors and
NP-complete to decide -colorability for every (and remains
intractable even for graphs of maximum degree 9 if ). On the other hand,
we prove positive results for -degenerate graphs with small , also
including planar graphs. Moreover we point out a fundamental connection with
the theory of the colorings of mixed hypergraphs. We list many open problems at
the end.Comment: 18 page
Generalisation : graphs and colourings
The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe
Entanglement and nonclassical properties of hypergraph states
Hypergraph states are multi-qubit states that form a subset of the locally
maximally entangleable states and a generalization of the well--established
notion of graph states. Mathematically, they can conveniently be described by a
hypergraph that indicates a possible generation procedure of these states;
alternatively, they can also be phrased in terms of a non-local stabilizer
formalism. In this paper, we explore the entanglement properties and
nonclassical features of hypergraph states. First, we identify the equivalence
classes under local unitary transformations for up to four qubits, as well as
important classes of five- and six-qubit states, and determine various
entanglement properties of these classes. Second, we present general conditions
under which the local unitary equivalence of hypergraph states can simply be
decided by considering a finite set of transformations with a clear
graph-theoretical interpretation. Finally, we consider the question whether
hypergraph states and their correlations can be used to reveal contradictions
with classical hidden variable theories. We demonstrate that various
noncontextuality inequalities and Bell inequalities can be derived for
hypergraph states.Comment: 29 pages, 5 figures, final versio
The Ising Partition Function: Zeros and Deterministic Approximation
We study the problem of approximating the partition function of the
ferromagnetic Ising model in graphs and hypergraphs. Our first result is a
deterministic approximation scheme (an FPTAS) for the partition function in
bounded degree graphs that is valid over the entire range of parameters
(the interaction) and (the external field), except for the case
(the "zero-field" case). A randomized algorithm (FPRAS)
for all graphs, and all , has long been known. Unlike most other
deterministic approximation algorithms for problems in statistical physics and
counting, our algorithm does not rely on the "decay of correlations" property.
Rather, we exploit and extend machinery developed recently by Barvinok, and
Patel and Regts, based on the location of the complex zeros of the partition
function, which can be seen as an algorithmic realization of the classical
Lee-Yang approach to phase transitions. Our approach extends to the more
general setting of the Ising model on hypergraphs of bounded degree and edge
size, where no previous algorithms (even randomized) were known for a wide
range of parameters. In order to achieve this extension, we establish a tight
version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a
classical result of Suzuki and Fisher.Comment: clarified presentation of combinatorial arguments, added new results
on optimality of univariate Lee-Yang theorem
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