Cycles through two edges in signed graphs

We give a characterization of when a signed graph $G$ with a pair of distinguished edges $e_1, e_2 \in E(G)$ has the property that all cycles containing both $e_1$ and $e_2$ have the same sign. This answers a question of Zaslavsky.Comment: 11 pages, 1 figur

Nowhere-zero 8-flows in cyclically 5-edge-connected, flow-admissible signed graphs

In 1983, Bouchet proved that every bidirected graph with a nowhere-zero integer-flow has a nowhere-zero 216-flow, and conjectured that 216 could be replaced with 6. This paper shows that for cyclically 5-edge-connected bidirected graphs that number can be replaced with 8.Comment: 14 page

Density of 33-critical signed graphs

We say that a signed graph is $k$-critical if it is not $k$-colorable but every one of its proper subgraphs is $k$-colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every $3$-critical signed graph on $n$ vertices has at least $\frac{3n-1}{2}$ edges, and that this bound is asymptotically tight. It follows that every signed planar or projective-planar graph of girth at least $6$ is (circular) $3$-colorable, and for the projective-planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph $C_{3}^*$, which is the positive triangle augmented with a negative loop on each vertex.Comment: 27 pages, 12 figure

Balanced-chromatic number and Hadwiger-like conjectures

Motivated by different characterizations of planar graphs and the 4-Color Theorem, several structural results concerning graphs of high chromatic number have been obtained. Toward strengthening some of these results, we consider the \emph{balanced chromatic number}, $\chi_b(\hat{G})$, of a signed graph $\hat{G}$. This is the minimum number of parts into which the vertices of a signed graph can be partitioned so that none of the parts induces a negative cycle. This extends the notion of the chromatic number of a graph since $\chi(G)=\chi_b(\tilde{G})$, where $\tilde{G}$ denotes the signed graph obtained from~$G$ by replacing each edge with a pair of (parallel) positive and negative edges. We introduce a signed version of Hadwiger's conjecture as follows. Conjecture: If a signed graph $\hat{G}$ has no negative loop and no $\tilde{K_t}$-minor, then its balanced chromatic number is at most $t-1$. We prove that this conjecture is, in fact, equivalent to Hadwiger's conjecture and show its relation to the Odd Hadwiger Conjecture. Motivated by these results, we also consider the relation between subdivisions and balanced chromatic number. We prove that if $(G, \sigma)$ has no negative loop and no $\tilde{K_t}$-subdivision, then it admits a balanced $\frac{79}{2}t^2$-coloring. This qualitatively generalizes a result of Kawarabayashi (2013) on totally odd subdivisions

Intracranial EEG fluctuates over months after implanting electrodes in human brain.

OBJECTIVE: Implanting subdural and penetrating electrodes in the brain causes acute trauma and inflammation that affect intracranial electroencephalographic (iEEG) recordings. This behavior and its potential impact on clinical decision-making and algorithms for implanted devices have not been assessed in detail. In this study we aim to characterize the temporal and spatial variability of continuous, prolonged human iEEG recordings. APPROACH: Intracranial electroencephalography from 15 patients with drug-refractory epilepsy, each implanted with 16 subdural electrodes and continuously monitored for an average of 18 months, was included in this study. Time and spectral domain features were computed each day for each channel for the duration of each patient\u27s recording. Metrics to capture post-implantation feature changes and inflexion points were computed on group and individual levels. A linear mixed model was used to characterize transient group-level changes in feature values post-implantation and independent linear models were used to describe individual variability. MAIN RESULTS: A significant decline in features important to seizure detection and prediction algorithms (mean line length, energy, and half-wave), as well as mean power in the Berger and high gamma bands, was observed in many patients over 100 d following implantation. In addition, spatial variability across electrodes declines post-implantation following a similar timeframe. All selected features decreased by 14-50% in the initial 75 d of recording on the group level, and at least one feature demonstrated this pattern in 13 of the 15 patients. Our findings indicate that iEEG signal features demonstrate increased variability following implantation, most notably in the weeks immediately post-implant. SIGNIFICANCE: These findings suggest that conclusions drawn from iEEG, both clinically and for research, should account for spatiotemporal signal variability and that properly assessing the iEEG in patients, depending upon the application, may require extended monitoring

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements