57 research outputs found
Online and quasi-online colorings of wedges and intervals
We consider proper online colorings of hypergraphs defined by geometric
regions. We prove that there is an online coloring algorithm that colors
intervals of the real line using colors such that for every
point , contained in at least intervals, not all the intervals
containing have the same color. We also prove the corresponding result
about online coloring a family of wedges (quadrants) in the plane that are the
translates of a given fixed wedge. These results contrast the results of the
first and third author showing that in the quasi-online setting 12 colors are
enough to color wedges (independent of and ). We also consider
quasi-online coloring of intervals. In all cases we present efficient coloring
algorithms
Positive co-degree density of hypergraphs
The minimum positive co-degree of a non-empty -graph , denoted
, is the maximum such that if is an -set
contained in a hyperedge of , then is contained in at least
distinct hyperedges of . Given a family of -graphs, we introduce
the {\it positive co-degree Tur\'an number} as the
maximum positive co-degree over all -vertex -graphs
that do not contain as a subhypergraph.
In this paper we concentrate on the behavior of for
-graphs . In particular, we determine asymptotics and bounds for several
well-known concrete -graphs (e.g.\ and the Fano plane).
We also show that, for -graphs, the limit ``jumps'' from to
, i.e., it never takes on values in the interval , and we
characterize which -graphs have . Our motivation comes
primarily from the study of (ordinary) co-degree Tur\'an numbers where a number
of results have been proved that inspire our results
Hierarchical graphs for rule-based modeling of biochemical systems
<p>Abstract</p> <p>Background</p> <p>In rule-based modeling, graphs are used to represent molecules: a colored vertex represents a component of a molecule, a vertex attribute represents the internal state of a component, and an edge represents a bond between components. Components of a molecule share the same color. Furthermore, graph-rewriting rules are used to represent molecular interactions. A rule that specifies addition (removal) of an edge represents a class of association (dissociation) reactions, and a rule that specifies a change of a vertex attribute represents a class of reactions that affect the internal state of a molecular component. A set of rules comprises an executable model that can be used to determine, through various means, the system-level dynamics of molecular interactions in a biochemical system.</p> <p>Results</p> <p>For purposes of model annotation, we propose the use of hierarchical graphs to represent structural relationships among components and subcomponents of molecules. We illustrate how hierarchical graphs can be used to naturally document the structural organization of the functional components and subcomponents of two proteins: the protein tyrosine kinase Lck and the T cell receptor (TCR) complex. We also show that computational methods developed for regular graphs can be applied to hierarchical graphs. In particular, we describe a generalization of Nauty, a graph isomorphism and canonical labeling algorithm. The generalized version of the Nauty procedure, which we call HNauty, can be used to assign canonical labels to hierarchical graphs or more generally to graphs with multiple edge types. The difference between the Nauty and HNauty procedures is minor, but for completeness, we provide an explanation of the entire HNauty algorithm.</p> <p>Conclusions</p> <p>Hierarchical graphs provide more intuitive formal representations of proteins and other structured molecules with multiple functional components than do the regular graphs of current languages for specifying rule-based models, such as the BioNetGen language (BNGL). Thus, the proposed use of hierarchical graphs should promote clarity and better understanding of rule-based models.</p
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