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Search results for: CHROMATIC NUMBER
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Dynamic coloring of graphs
PublicationDynamics is an inherent feature of many real life systems so it is natural to define and investigate the properties of models that reflect their dynamic nature. Dynamic graph colorings can be naturally applied in system modeling, e.g. for scheduling threads of parallel programs, time sharing in wireless networks, session scheduling in high-speed LAN's, channel assignment in WDM optical networks as well as traffic scheduling. In...
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Scheduling of unit-length jobs with cubic incompatibility graphs on three uniform machines
PublicationWe consider the problem of scheduling n identical jobs on 3 uniform machines with speeds s1, s2, and s3 to minimize the schedule length. We assume that jobs are subject to some kind of mutual exclusion constraints, modeled by a cubic incompatibility graph. We how that if the graph is 2-chromatic then the problem can be solved in O(n^2) time. If the graph is 3-chromatic, the problem becomes NP-hard even if s1>s2=s3.
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Relations between the domination parameters and the chromatic index of a graph
PublicationIn this paper we show bounds for the sum and the product of the domination parameters and the chromatic index of a graph. We alsopresent some families of graphs for which these bounds are achieved.
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Interval Edge Coloring of Bipartite Graphs with Small Vertex Degrees
PublicationAn edge coloring of a graph G is called interval edge coloring if for each v ∈ V(G) the set of colors on edges incident to v forms an interval of integers. A graph G is interval colorable if there is an interval coloring of G. For an interval colorable graph G, by the interval chromatic index of G, denoted by χ'_i(G), we mean the smallest number k such that G is interval colorable with k colors. A bipartite graph G is called (α,β)-biregular...
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Edge coloring of graphs of signed class 1 and 2
PublicationRecently, Behr (2020) introduced a notion of the chromatic index of signed graphs and proved that for every signed graph (G, σ) it holds that ∆(G) ≤ χ′(G,σ) ≤ ∆(G) + 1, where ∆(G) is the maximum degree of G and χ′ denotes its chromatic index. In general, the chromatic index of (G, σ) depends on both the underlying graph G and the signature σ. In the paper we study graphs G for which χ′(G, σ) does not depend on σ. To this aim we...
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A graph coloring approach to scheduling of multiprocessor tasks on dedicated machines with availability constraints
PublicationWe address a generalization of the classical 1- and 2-processor unit execution time scheduling problem on dedicated machines. In our chromatic model of scheduling machines have non-simultaneous availability times and tasks have arbitrary release times and due dates. Also, the versatility of our approach makes it possible to generalize all known classical criteria of optimality. Under these stipulations we show that the problem...
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A note on polynomial algorithm for cost coloring of bipartite graphs with Δ ≤ 4
PublicationIn the note we consider vertex coloring of a graph in which each color has an associated cost which is incurred each time the color is assigned to a vertex. The cost of coloring is the sum of costs incurred at each vertex. We show that the minimum cost coloring problem for n-vertex bipartite graph of degree ∆≤4 can be solved in O(n^2) time. This extends Jansen’s result [K.Jansen,The optimum cost chromatic partition problem, in:...
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Detection and segmentation of moving vehicles and trains using Gaussian mixtures, shadow detection and morphological processing
PublicationSolution presented in this paper combines background modelling, shadow detection and morphological and temporal processing into one system responsible for detection and segmentation of moving objects recorded with a static camera. Vehicles and trains are detected based on their pixellevel difference from the continually updated background model utilizing a Gaussian mixture calculated separately for every pixel. The shadow detection...
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Zirconia ceramics with additions of Alumina for advanced tribological and biomedical applications
PublicationThe results of an investigation on slip cast and sintered Y2O3 (3 wt%)- stabilized ZrO2 with additions of 5, 10, 15 wt% Al2O3 are reported. The surface roughness, porosity and density of the samples were measured. The hardness HRc and Hv, fracture toughness K1C, and friction coefficients were also measured using standard methods. The structural properties of the samples were observed by Scanning Electron Microscopy (SEM). The surface...
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Nonlinearity shaping in nanostructured glass-diamond hybrid materials for optical fiber preforms
PublicationNanodiamond integration with optical fibers has proved a compelling methodology for magneto-optics. We reveal that the applicability of nanodiamonds in nonlinear optics goes beyond the previous demonstrations of frequency converters. Instead, we exploit the recently reported volumetric integration of nanodiamonds along the optical fiber core and show that the nonlinear response of glasses can be manipulated by nanodiamonds. By...