38200007
Total colorings of equibipartite graphs
Abstract The total chromatic number χτ ( G ) of a graph G is the least number of colors needed to color the vertices and the edges of G such that no adjacent or incident pair of elements receive the same color. A simple graph G is called type 1 if χτ ( G ) = Δ ( G ) + 1, where Δ ( G ) is the maximum degree of G . In this paper we prove the following conjecture of Chen et al.: An ( n − 2)-regular equibipartite graph K n , n − E ( J ) is type 1 if and only if J contains a 4-cycle.
32727998
Equitable Coloring of Trees
Abstract A graph is equitably k -colorable if its vertices can be partitioned into k independent sets of as near equal sizes as possible. Regarding a non-null tree T as a bipartite graph T ( X , Y ), we show that T is equitably k -colorable if and only if (i) k ≥ 2 when | | X | − | Y | | ≤ 1; (ii) k ≥ max{3, ⌈(| T | + 1)/(α( T − N [ v ]) + 2)⌉} when | | X | − | Y | | > 1. In case (ii), v is an arbitrary vertex of maximum degree in T and the number α( T − N [ v ]) denotes the independence number of the subgraph of T obtained by deleting v and all its adjacent vertices.
42441601
Hamiltonian uniform subset graphs
The uniform subset graph G(n, k, t) is defined to have all k-subsets of an n-set as vertices and edges joining k-subsets intersecting at t elements. We conjecture that G(n, k, t) is hamiltonian when it is different from the Petersen graph and does possess cycles. We verify this conjecture for k − t = 1, 2, 3 and for suitably large n when t = 0, 1.
4020261
On-Line Scheduling a Batch Processing System to Minimize Total Weighted Job Completion Time
Scheduling a batch processing system has been extensively studied in the last decade. A batch processing system is modelled as a machine that can process up to b jobs simultaneously as a batch. The scheduling problem involves assigning all n jobs to batches and determining the batch sequence in such a way that certain objective function of job completion times Cj is minimized. In this paper, we address the scheduling problem under the on-line setting in the sense that we construct our schedule irrevocably as time proceeds and do not know of the existence of any job that may arrive later. Our objective is to minimize the total weighted completion time ∑wjCj. We provide a linear time on-line algorithm for the unrestrictive model (i.e., b ≥ n) and show that the algorithm is 10/3-competitive. For the restrictive model (i.e., b < n), we first consider the (off-line) problem of finding a maximum independent vertex set in an interval graph with cost constraint (MISCP), which is NP-hard. We give a dual fully polynomial time approximation scheme for MISCP, which leads us to a (4 + ∈)-competitive on-line algorithm for any ∈ > 0 for the original on-line scheduling problem. These two on-line algorithms are the first deterministic algorithms of constant performance guarantees.
5893940
Multiresolution tetrahedral framework for visualizing regular volume data
The authors present a multiresolution framework, called Multi-Tetra framework, that approximates volume data with different levels-of-detail tetrahedra. The framework is generated through a recursive subdivision of the volume data and is represented by binary trees. Instead of using a certain level of the Multi-Tetra framework for approximation, an error-based model (EBM) is generated by recursively fusing a sequence of tetrahedra from different levels of the Multi-Tetra framework. The EBM significantly reduces the number of voxels required to model an object, while preserving the original topology. The approach provides continuous distribution of rendered intensity or generated isosurfaces along boundaries of different levels-of-detail thus solving the crack problem. The model supports typical rendering approaches, such as marching cubes, direct volume projection, and splatting. Experimental results demonstrate the strengths of the approach.
13073706
Graph homotopy and Graham homotopy
Abstract Simple-homotopy for cell complexes is a special type of topological homotopy constructed by elementary collapses and elementary expansions. In this paper, we introduce graph homotopy for graphs and Graham homotopy for hypergraphs and study the relation between the two homotopies and the simple-homotopy for cell complexes. The graph homotopy is useful to describe topological properties of discretized geometric figures, while the Graham homotopy is essential to characterize acyclic hypergraphs and acyclic relational database schemes.
4000197
Lot-sizing scheduling with batch setup times
This paper is concerned with scheduling independent jobs on m parallel machines in such a way that the makespan is minimized. Each job j is allowed to split arbitrarily into several parts, which can be individually processed on any machine at any time. However, a setup for uninterrupted sj time units is required before any split part of job j can be processed on any machine. The problem is strongly NP-hard if the number m of machines is variable and weakly NP-hard otherwise. We give a polynomial-time $$\frac{5}{3}$$-approximation algorithm for the former case and a fully polynomial-time approximation scheme for the latter.
1980658
Geometric Sieving: Automated Distributed Optimization of 3D Motifs for Protein Function Prediction
Determining the function of all proteins is a recurring theme in modern biology and medicine, but the sheer number of proteins makes experimental approaches impractical. For this reason, current efforts have considered in silico function prediction in order to guide and accelerate the function determination process. One approach to predicting protein function is to search functionally uncharacterized protein structures (targets), for substructures with geometric and chemical similarity (matches), to known active sites (motifs). Finding a match can imply that the target has an active site similar to the motif, suggesting functional homology. 
 
An effective function predictor requires effective motifs – motifs whose geometric and chemical characteristics are detected by comparison algorithms within functionally homologous targets (sensitive motifs), which also are not detected within functionally unrelated targets (specific motifs). Designing effective motifs is a difficult open problem. Current approaches select and combine structural, physical, and evolutionary properties to design motifs that mirror functional characteristics of active sites. 
 
We present a new approach, Geometric Sieving (GS), which refines candidate motifs into optimized motifs with maximal geometric and chemical dissimilarity from all known protein structures. The paper discusses both the usefulness and the efficiency of GS. We show that candidate motifs from six well-studied proteins, including α-Chymotrypsin, Dihydrofolate Reductase, and Lysozyme, can be optimized with GS to motifs that are among the most sensitive and specific motifs possible for the candidate motifs. For the same proteins, we also report results that relate evolutionarily important motifs with motifs that exhibit maximal geometric and chemical dissimilarity from all known protein structures. Our current observations show that GS is a powerful tool that can complement existing work on motif design and protein function prediction.