Generalized Colorings of Graphs

A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique

Acyclic homomorphisms to stars of graph Cartesian products and chordal bipartite graphs

AbstractHomomorphisms to a given graph H (H-colourings) are considered in the literature among other graph colouring concepts. We restrict our attention to a special class of H-colourings, namely H is assumed to be a star. Our additional requirement is that the set of vertices of a graph G mapped into the central vertex of the star and any other colour class induce in G an acyclic subgraph. We investigate the existence of such a homomorphism to a star of given order. The complexity of this problem is studied. Moreover, the smallest order of a star for which a homomorphism of a given graph G with desired features exists is considered. Some exact values and many bounds of this number for chordal bipartite graphs, cylinders, grids, in particular hypercubes, are given. As an application of these results, we obtain some bounds on the cardinality of the minimum feedback vertex set for specified graph classes

Gene Alterations of Ovarian Cancer Cells Expressing Estrogen Receptors by Estrogen and Bisphenol A Using Microarray Analysis

Since endocrine disrupting chemicals (EDCs) may interfere with the endocrine system(s) of our body and have an estrogenicity, we evaluated the effect(s) of bisphenol A (BPA) on the transcriptional levels of altered genes in estrogen receptor (ER)-positive BG-1 ovarian cancer cells by microarray and real-time polymerase-chain reaction. In this study, treatment with 17β-estradiol (E2) or BPA increased mRNA levels of E2-responsive genes related to apoptosis, cancer and cell cycle, signal transduction and nucleic acid binding etc. In parallel with their microarray data, the mRNA levels of some altered genes including RAB31_MEMBER RAS ONCOGENE FAMILY (U59877), CYCLIN D1 (X59798), CYCLIN-DEPENDENT KINASE 4 (U37022), IGF-BINDING PROTEIN 4 (U20982), and ANTI-MULLERIAN HORMONE (NM_000479) were significantly induced by E2 or BPA in this cell model. These results indicate that BPA in parallel with E2 induced the transcriptional levels of E2-responsive genes in an estrogen receptor (ER)-positive BG-1 cells. In conclusion, these microarray and real-time polymerase-chain reaction results indicate that BPA, a potential weak estrogen, may have estrogenic effect by regulating E2-responsive genes in ER-positive BG-1 cells and BG-1 cells would be the best in vitro model to detect these estrogenic EDCs

A generic algorithm for layout of biological networks

BackgroundBiological networks are widely used to represent processes in biological systems and to capture interactions and dependencies between biological entities. Their size and complexity is steadily increasing due to the ongoing growth of knowledge in the life sciences. To aid understanding of biological networks several algorithms for laying out and graphically representing networks and network analysis results have been developed. However, current algorithms are specialized to particular layout styles and therefore different algorithms are required for each kind of network and/or style of layout. This increases implementation effort and means that new algorithms must be developed for new layout styles. Furthermore, additional effort is necessary to compose different layout conventions in the same diagram. Also the user cannot usually customize the placement of nodes to tailor the layout to their particular need or task and there is little support for interactive network exploration.ResultsWe present a novel algorithm to visualize different biological networks and network analysis results in meaningful ways depending on network types and analysis outcome. Our method is based on constrained graph layout and we demonstrate how it can handle the drawing conventions used in biological networks.ConclusionThe presented algorithm offers the ability to produce many of the fundamental popular drawing styles while allowing the exibility of constraints to further tailor these layouts.publishe