If S=(s1,s2,…) is a non-decreasing sequence of positive integers, then the S-packing k-coloring of a graph G is a mapping c:V(G)→k such that if c(u)=c(v)=i for u≠v∈V(G), then dG(u,v)>si. The ...S-packing chromatic number of G is the smallest integer k such that G admits an S-packing k-coloring. A graph G is χS-vertex-critical if χS(G−u)<χS(G) for each u∈V(G). If G is χS-vertex-critical and χS(G)=k, then G is k−χS-vertex-critical. In this paper, 4−χS-vertex-critical graphs are characterized for sequences S=(1,s2,s3,…) with s2≥3. There are 28 sporadic examples and two infinite families of such graphs.
We study the problem of finding a maximum-cardinality set of r-cliques in an undirected graph of fixed maximum degree Δ, subject to the cliques in that set being either vertex disjoint or edge ...disjoint. It is known for r=3 that the vertex-disjoint (edge-disjoint) problem is solvable in linear time if Δ=3 (Δ=4) but APX-hard if Δ≥4 (Δ≥5).
We generalise these results to an arbitrary but fixed r≥3, and provide a complete complexity classification for both the vertex- and edge-disjoint variants in graphs of maximum degree Δ.
Specifically, we show that the vertex-disjoint problem is solvable in linear time if Δ<3r/2−1, solvable in polynomial time if Δ<5r/3−1, and APX-hard if Δ≥⌈5r/3⌉−1. We also show that if r≥6 then the above implications also hold for the edge-disjoint problem. If r≤5, then the edge-disjoint problem is solvable in linear time if Δ<3r/2−1, solvable in polynomial time if Δ≤2r−2, and APX-hard if Δ>2r−2.
In this work, the hydroxypropyl-β-cyclodextrin (HP-β-CD)/Epicatechin (EC) clathrate compounds were rapidly prepared based on an ultrasound-mediated method, and Polycaprolactone (PCL)/Locust bean gum ...(LBG) nanofibers loaded clathrate compounds were fabricated by electrostatic spinning (ELS) for fruit packaging. The results of infrared spectrum and crystal type analysis proved that clathrate compounds were successfully prepared. With the addition of clathrate compounds, the diameter of fibers increased from 553.43 to 1273.47 nm, and hydrogen bonds were formed between clathrate compounds and fibrous membranes, which improved the thermal stability, reduced the crystallinity, and enhanced the hydrophilicity and gas permeability of fibrous membranes. The fibrous membranes indicated sustained release of EC for 240 h, retaining the activity of EC and demonstrating good bacteriostatic ability in vitro and in vivo. The test results showed that the antibacterial fibrous membranes prepared in this work have a positive application prospect for fruit packaging.
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Densification of cubic particles under three-dimensional (3D) mechanical vibration was studied experimentally. Effects of vibration time (t), frequency (f), vibration amplitude (A), vibration ...acceleration (Г), container size (D) and particle sphericity (φ) on packing density (ρ) were comprehensively analyzed. To identify the effects of particle shape, the packing densification of cuboid 1 (12mm×12mm×28mm) and cuboid 2 (4mm×8mm×16mm) were systematically studied under the same conditions. The results show that the structure of cubic particles can be first densified from random loose packing (RLP) to random close packing (RCP) and then to ordered packing (OP) gradually. In comparison, cuboid 1 and cuboid 2 particles can only form RCP structure, but cannot achieve OP even under further vibration. Vibration parameters (t, f, A and Г) are shown to be important to the packing densification. Based on the results of varying f and A, A - f phase diagrams are established for choosing the optimal vibration parameters to achieve the desired dense packing structures. Besides, it is shown the size of container (wall effect) has monotonic influence on packing density, i.e., the larger container size corresponds to the less wall effect and higher packing density. Cubic particles can form ordered packing because of the geometrical symmetry with aspect ratio of 1. After eliminating the wall effect by extrapolating the packing densities in different sized containers, typical packing densities of the three types of particles are obtained with ρRLP=0.658, ρRCP=0.830 and ρOP=0.965 for cubic particles; ρRLP=0.625 and ρRCP=0.740 for cuboid 1 particles; ρRLP=0.591 and ρRCP=0.731 for cuboid 2 particles. These findings indicate cubic particles are efficient in densification which can be readily realized through proper 3D mechanical vibration.
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•Packing densification of cubic and cuboid particles subjected to 3D mechanical vibration was studied experimentally.•The influences of vibration conditions on the packing density were analyzed.•The transition from random packing to ordered packing was realized for cubic particles.•Critical packing densities of the three types of particles were obtained after eliminating the wall effect.
In online three-dimensional packing problems (3D-PPs), unlike offline problems, items arrive sequentially and require immediate packing decisions without any information about the quantities and ...sizes of the items to come. Heuristic methods are of great importance in solving online problems to find good solutions in a reasonable amount of time. However, the literature on heuristics for online problems is sparse. As our first contribution, we developed a pool of heuristics applicable to online 3D-PPs with complementary performance on different sets of instances. Computational results showed that in terms of the number of used bins, in all problem instances, at least one of our heuristics had a better or equal performance compared to existing heuristics in the literature. The developed heuristics are also fully applicable to an intermediate class between offline and online problems, referred to in this paper as a specific type of “semi-online with full look-ahead”, which has several practical applications. In this class, as in offline problems, complete information about all items is known in advance (i.e., full look-ahead); however, due to time or space constraints, as in online problems, items should be packed immediately in the order of their arrival. As our second contribution, we presented an algorithm selection framework, building on developed heuristics and utilizing prior information about items in this specific class of problems. We used supervised machine learning techniques to find the relationship between the features of problem instances and the performance of heuristics and to build a prediction model. The results indicate an 88% accuracy in predicting (identifying) the most promising heuristic(s) for solving any new instance from this class of problems.
•Several heuristic methods to solve online 3D-packing problems (3D-PPs) are proposed.•Proposed heuristics are applicable to semi-online 3D-PPs with full look-ahead.•An algorithm selection framework to predict the most promising heuristic is proposed.•The prediction performance of three learning algorithms is compared in the framework.•The multi-label neural network demonstrates a notably high prediction accuracy.
The structure of mono-sized sphere packings (diameter d) in cylindrical containers (diameter D and height H) both with and without inner cylinders (diameter Di) has been investigated in detail by ...means of advanced X-ray computed tomography. The geometrical parameters were varied in a wide range; in all experiments 1d vertical vibration was applied. Five experiments were selected with characteristically differing local packing structures. The influence of container geometry, filling and vibration procedures on the formation of regular packings is discussed and a simple correlation is presented to assess whether structured packings occupy a significant fraction of the total packed volume.
For a packing with moderate densification, the regular structures are restricted to small wall zones and a random packing exists in the largest part of the packing volume. By selecting appropriate vibration parameters, the zones with regular structures can increase considerably and can persist in the total packed volume. The increasing crystallisation causes an increase of the container packing fraction. For cylinders with H/D ≫1 and moderate D/d, regular structures develop preferentially in radial direction from a hexagonal layer at the concave wall. For H/D<1 and D/d≫1, hexagonal dense structures grow preferentially above the flat bottom plate and can occupy a great portion of the total volume. The role of granular convection on these crystallisation processes has been addressed. Previous statements that the thickness of wall zones is ≈(4–5)d are not generally valid for mono-sized sphere packings; the development of a comprehensive correlation is the task of a future work.
Structural details of the packings close to concave, plane and convex walls are analysed via void fraction distributions, sphere centre positions, contact angle distributions, coordination numbers, radial distribution function and Voronoi tessellation. The combination of these methods provides a comprehensive understanding of structural details. Only a few characteristic results are presented; special topics will be the subject of forthcoming publications.
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•Regular packing structures induced by walls are analysed.•Different types of ordered structures (crystallisation) are observed.•Various measures are applied to quantify order.•The combination of these methods enlarges the understanding of structures.
•High packing achieved within a wide range of aggregate mix combinations.•Compressible Packing Model valid for both limestone and EAFS aggregates.•The 3-Parameter Particle Packing Model not useful ...for EAFS aggregates.
An experimental method and two discrete Particle Packing Models are assessed for aggregate mix design. The aim is to model the most compacted structure of commercial natural limestone aggregate and electric steelmaking slag fractions. The experimental method is reliable and effective, although somewhat laborious. Regarding the discrete models, the Compressible Packing Model and the 3-Parameter Particle Packing Model showed high reliability when used to perform the most compact of both the ternary and the quaternary aggregate mixtures. Eight concrete mixes were manufactured to produce concretes of low and medium workability, containing the highest possible content by volume of coarse aggregate, and their resultant in-fresh properties, defined through slump and density, confirmed the validity of using the Particle Packing Models for each mix dosage under study.
Metakaolin (MK) and silica fume (SF) have been added either individually or in combination to improve concrete performance. Recent studies proved that the combined addition of MK and SF offers ...synergistic effects, but the mechanism behind is still not unclear. Herein, it is postulated that the synergistic effects arise mainly from the successive filling effects of the MK and SF into voids between larger size particles, which increase the wet packing density (WPD). The present work indicated that the slurry formed of the water and ultrafine particles plays an important role. Good correlation of the workability measures to the slurry film thickness (SFT) revealed the governing factor of SFT for the first time. Moreover, the synergistic effects of MK and SF on the compressive and flexural strengths have been studied based on cementing efficiency and synergistic factors. Correlation of the synergistic factor to the WPD revealed that the WPD plays an important role.
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•The simultaneous use of metakaolin and silica fume is explored.•The concept of slurry film thickness is established for modeling.•Formulas based on slurry film thickness are derived for workability.•Synergistic factor and particle packing are associated with strengths.
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