105 research outputs found
On the density of sets of the Euclidean plane avoiding distance 1
A subset is said to avoid distance if: In this paper we study the number
which is the supremum of the upper densities of measurable
sets avoiding distance 1 in the Euclidean plane. Intuitively, represents the highest proportion of the plane that can be filled by a
set avoiding distance 1. This parameter is related to the fractional chromatic
number of the plane.
We establish that and .Comment: 11 pages, 5 figure
Partitionable graphs arising from near-factorizations of finite groups
AbstractIn 1979, two constructions for making partitionable graphs were introduced in (by Chvátal et al. (Ann. Discrete Math. 21 (1984) 197)). The graphs produced by the second construction are called CGPW graphs. A near-factorization (A,B) of a finite group is roughly speaking a non-trivial factorization of G minus one element into two subsets A and B. Every CGPW graph with n vertices turns out to be a Cayley graph of the cyclic group Zn, with connection set (A−A)⧹{0}, for a near-factorization (A,B) of Zn. Since a counter-example to the Strong Perfect Graph Conjecture would be a partitionable graph (Padberg, Math. Programming 6 (1974) 180), any ‘new’ construction for making partitionable graphs is of interest. In this paper, we investigate the near-factorizations of finite groups in general, and their associated Cayley graphs which are all partitionable. In particular, we show that near-factorizations of the dihedral groups produce every CGPW graph of even order. We present some results about near-factorizations of finite groups which imply that a finite abelian group with a near-factorization (A,B) such that |A|⩽4 must be cyclic (already proved by De Caen et al. (Ars Combin. 29 (1990) 53)). One of these results may be used to speed up exhaustive calculations. At last, we prove that there is no counter-example to the Strong Perfect Graph Conjecture arising from near-factorizations of a finite abelian group of even order
On the theta number of powers of cycle graphs
We give a closed formula for Lovasz theta number of the powers of cycle
graphs and of their complements, the circular complete graphs. As a
consequence, we establish that the circular chromatic number of a circular
perfect graph is computable in polynomial time. We also derive an asymptotic
estimate for this theta number.Comment: 17 page
On the density of sets avoiding parallelohedron distance 1
The maximal density of a measurable subset of R^n avoiding Euclidean
distance1 is unknown except in the trivial case of dimension 1. In this paper,
we consider thecase of a distance associated to a polytope that tiles space,
where it is likely that the setsavoiding distance 1 are of maximal density
2^-n, as conjectured by Bachoc and Robins. We prove that this is true for n =
2, and for the Vorono\"i regions of the lattices An, n >= 2
How unique is Lovász's theta function?
International audienceThe famous Lovász's ϑ function is computable in polynomial time for every graph, as a semi-definite program (Grötschel, Lovász and Schrijver, 1981). The chromatic number and the clique number of every perfect graph G are computable in polynomial time. Despite numerous efforts since the last three decades, stimulated by the Strong Perfect Graph Theo-rem (Chudnovsky, Robertson, Seymour and Thomas, 2006), no combinatorial proof of this result is known. In this work, we try to understand why the "key properties" of Lovász's ϑ function make it so "unique". We introduce an infinite set of convex functions, which includes the clique number ω and ϑ . This set includes a sequence of linear programs which are monotone increasing and converging to ϑ . We provide some evidences that ϑ is the unique function in this setting allowing to compute the chromatic number of perfect graphs in polynomial time
Consecutive ones matrices for multi-dimensional orthogonal packing problems
International audienceThe multi-dimensional orthogonal packing problem (OPP) is a well studied decisional problem. Given a set of items with rectangular shapes, the problem is to decide whether there is a non-overlapping packing of these items in a rectangular bin. The rotation of items is not allowed. A powerful caracterization of packing configurations by means of interval graphs was recently introduced. In this paper, we propose a new algorithm using consecutive ones matrices as data structure. This new algorithm is then used to solve the two-dimensional orthogonal knapsack problem. Computational results are reported, which show its effectiveness
On the density of sets of the Euclidean plane avoiding distance
International audienc
Exhumation, crustal deformation, and thermal structure of the Nepal Himalaya derived from the inversion of thermochronological and thermobarometric data and modeling of the topography
Two end‐member kinematic models of crustal shortening across the Himalaya are
currently debated: one assumes localized thrusting along a single major thrust fault, the
Main Himalayan Thrust (MHT) with nonuniform underplating due to duplexing, and the
other advocates for out‐of‐sequence (OOS) thrusting in addition to thrusting along the
MHT and underplating. We assess these two models based on the modeling of
thermochronological, thermometric, and thermobarometric data from the central Nepal
Himalaya. We complement a data set compiled from the literature with 114 ^(40)Ar/^(39)Ar,
10 apatite fission track, and 5 zircon (U‐Th)/He thermochronological data. The data are
predicted using a thermokinematic model (PECUBE), and the model parameters are
constrained using an inverse approach based on the Neighborhood Algorithm. The model
parameters include geometric characteristics as well as overthrusting rates, radiogenic heat
production in the High Himalayan Crystalline (HHC) sequence, the age of initiation of
the duplex or of out-of-sequence thrusting. Both models can provide a satisfactory fit to the
inverted data. However, the model with out-of-sequence thrusting implies an unrealistic
convergence rate ≥30 mm yr^(−1). The out-of-sequence thrust model can be adjusted to fit the
convergence rate and the thermochronological data if the Main Central Thrust zone is
assigned a constant geometry and a dip angle of about 30° and a slip rate of <1 mm yr^(−1). In
the duplex model, the 20 mm yr^(−1) convergence rate is partitioned between an overthrusting
rate of 5.8 ± 1.4 mm yr^(−1) and an underthrusting rate of 14.2 ± 1.8 mm yr^(−1). Modern rock
uplift rates are estimated to increase from about 0.9 ± 0.31 mm yr^(−1) in the Lesser Himalaya to
3.0 ± 0.9 mm yr^(−1) at the front of the high range, 86 ± 13 km from the Main Frontal Thrust.
The effective friction coefficient is estimated to be 0.07 or smaller, and the radiogenic
heat production of HHC units is estimated to be 2.2 ± 0.1 µWm^(−3). The midcrustal
duplex initiated at 9.8 ± 1.7 Ma, leading to an increase of uplift rate at front of the High
Himalaya from 0.9 ± 0.31 to 3.05 ± 0.9 mm yr^(−1). We also run 3-D models by coupling
PECUBE with a landscape evolution model (CASCADE). This modeling shows that the
effect of the evolving topography can explain a fraction of the scatter observed in the data but
not all of it, suggesting that lateral variations of the kinematics of crustal deformation and
exhumation are likely. It has been argued that the steep physiographic transition at the foot of
the Greater Himalayan Sequence indicates OOS thrusting, but our results demonstrate
that the best fit duplex model derived from the thermochronological and thermobarometric
data reproduces the present morphology of the Nepal Himalaya equally well
On the Lovasz's Theta function of power of chordless cycles
International audienc
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