Online Multi-Coloring with Advice

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c

Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices

Let S be a flat surface of genus g with cone type singularities. Given a bipartite graph G isoradially embedded in S, we define discrete analogs of the 2^{2g} Dirac operators on S. These discrete objects are then shown to converge to the continuous ones, in some appropriate sense. Finally, we obtain necessary and sufficient conditions on the pair (S,G) for these discrete Dirac operators to be Kasteleyn matrices of the graph G. As a consequence, if these conditions are met, the partition function of the dimer model on G can be explicitly written as an alternating sum of the determinants of these 2^{2g} discrete Dirac operators.Comment: 39 pages, minor change

Hypercellular graphs: partial cubes without Q3−Q_3^- as partial cube minor

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -- a property naturally generalizing the notion of median graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier draft (Figure 6.

Bipartite partial duals and circuits in medial graphs

It is well known that a plane graph is Eulerian if and only if its geometric dual is bipartite. We extend this result to partial duals of plane graphs. We then characterize all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph.Comment: v2: minor changes. To appear in Combinatoric

From data towards knowledge: Revealing the architecture of signaling systems by unifying knowledge mining and data mining of systematic perturbation data

Genetic and pharmacological perturbation experiments, such as deleting a gene and monitoring gene expression responses, are powerful tools for studying cellular signal transduction pathways. However, it remains a challenge to automatically derive knowledge of a cellular signaling system at a conceptual level from systematic perturbation-response data. In this study, we explored a framework that unifies knowledge mining and data mining approaches towards the goal. The framework consists of the following automated processes: 1) applying an ontology-driven knowledge mining approach to identify functional modules among the genes responding to a perturbation in order to reveal potential signals affected by the perturbation; 2) applying a graph-based data mining approach to search for perturbations that affect a common signal with respect to a functional module, and 3) revealing the architecture of a signaling system organize signaling units into a hierarchy based on their relationships. Applying this framework to a compendium of yeast perturbation-response data, we have successfully recovered many well-known signal transduction pathways; in addition, our analysis have led to many hypotheses regarding the yeast signal transduction system; finally, our analysis automatically organized perturbed genes as a graph reflecting the architect of the yeast signaling system. Importantly, this framework transformed molecular findings from a gene level to a conceptual level, which readily can be translated into computable knowledge in the form of rules regarding the yeast signaling system, such as "if genes involved in MAPK signaling are perturbed, genes involved in pheromone responses will be differentially expressed"