On vertex coloring without monochromatic triangles

We study a certain relaxation of the classic vertex coloring problem, namely, a coloring of vertices of undirected, simple graphs, such that there are no monochromatic triangles. We give the first classification of the problem in terms of classic and parametrized algorithms. Several computational complexity results are also presented, which improve on the previous results found in the literature. We propose the new structural parameter for undirected, simple graphs -- the triangle-free chromatic number $\chi_3$. We bound $\chi_3$ by other known structural parameters. We also present two classes of graphs with interesting coloring properties, that play pivotal role in proving useful observation about our problem. We give/ask several conjectures/questions throughout this paper to encourage new research in the area of graph coloring.Comment: Extended abstrac

Antarctic Surface Reflectivity Measurements from the ANITA-3 and HiCal-1 Experiments

The primary science goal of the NASA-sponsored ANITA project is measurement of ultra-high energy neutrinos and cosmic rays, observed via radio-frequency signals resulting from a neutrino- or cosmic ray- interaction with terrestrial matter (atmospheric or ice molecules, e.g.). Accurate inference of the energies of these cosmic rays requires understanding the transmission/reflection of radio wave signals across the ice-air boundary. Satellite-based measurements of Antarctic surface reflectivity, using a co-located transmitter and receiver, have been performed more-or-less continuously for the last few decades. Satellite-based reflectivity surveys, at frequencies ranging from 2--45 GHz and at near-normal incidence, yield generally consistent reflectivity maps across Antarctica. Using the Sun as an RF source, and the ANITA-3 balloon borne radio-frequency antenna array as the RF receiver, we have also measured the surface reflectivity over the interval 200-1000 MHz, at elevation angles of 12-30 degrees, finding agreement with the Fresnel equations within systematic errors. To probe low incidence angles, inaccessible to the Antarctic Solar technique and not probed by previous satellite surveys, a novel experimental approach ("HiCal-1") was devised. Unlike previous measurements, HiCal-ANITA constitute a bi-static transmitter-receiver pair separated by hundreds of kilometers. Data taken with HiCal, between 200--600 MHz shows a significant departure from the Fresnel equations, constant with frequency over that band, with the deficit increasing with obliquity of incidence, which we attribute to the combined effects of possible surface roughness, surface grain effects, radar clutter and/or shadowing of the reflection zone due to Earth curvature effects.Comment: updated to match publication versio

On Coloring Resilient Graphs

We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NP-hardness and the existence of efficient algorithms for resilient instances. In particular, we study $r$-resiliently $k$-colorable graphs, which are those $k$-colorable graphs that remain $k$-colorable even after the addition of any $r$ new edges. We prove lower bounds on the NP-hardness of coloring resiliently colorable graphs, and provide an algorithm that colors sufficiently resilient graphs. We also analyze the corresponding notion of resilience for $k$-SAT. This notion of resilience suggests an array of open questions for graph coloring and other combinatorial problems.Comment: Appearing in MFCS 201

Regulation and drug modulation of a voltage-gated sodium channel: Pivotal role of the S4–S5 linker in activation and slow inactivation

Voltage-gated sodium (NaV) channels control excitable cell functions. While structural investigations have revealed conformation details of different functional states, the mechanisms of both activation and slow inactivation remain unclear. Here, we identify residue T140 in the S4–S5 linker of the bacterial voltage-gated sodium channel NaChBac as critical for channel activation and drug effects on inactivation. Mutations at T140 either attenuate activation or render the channel nonfunctional. Propofol, a clinical anesthetic known to inhibit NaChBac by promoting slow inactivation, binds to a pocket between the S4–S5 linker and S6 helix in a conformation-dependent manner. Using 19F-NMR to quantify site-specific binding by saturation transfer differences (STDs), we found strong STDs in inactivated, but not activated, NaChBac. Molecular dynamics simulations show a highly dynamic pocket in the activated conformation, limiting STD buildup. In contrast, drug binding to this pocket promotes and stabilizes the inactivated states. Our results provide direct experimental evidence showing distinctly different associations between the S4–S5 linker and S6 helix in activated and inactivated states. Specifically, an exchange occurs between interaction partners T140 and N234 of the same subunit in activation, and T140 and N225 of the domain-swapped subunit in slow inactivation. The drug action on slow inactivation of prokaryotic NaV channels seems to have a mechanism similar to the recently proposed “door-wedge” action of the isoleucine-phenylalanine-methionine (IFM) motif on the fast inactivation of eukaryotic NaV channels. Elucidating this gating mechanism points to a possible direction for conformation-dependent drug development