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1. Phasic cholinergic signaling promotes emergence of local gamma rhythms in excitatory–inhibitory networks

   https://doi.org/10.1111/ejn.14744  

   Lu, Yiqing; Sarter, Martin; Zochowski, Michal; Booth, Victoria (May 2020, European Journal of Neuroscience)

   Abstract Recent experimental results have shown that the detection of cues in behavioral attention tasks relies on transient increases of acetylcholine (ACh) release in frontal cortex and cholinergically driven oscillatory activity in the gamma frequency band (Howe et al. Journal of Neuroscience, 2017, 37, 3215). The cue‐induced gamma rhythmic activity requires stimulation of M1 muscarinic receptors. Using biophysical computational modeling, we show that a network of excitatory (E) and inhibitory (I) neurons that initially displays asynchronous firing can generate transient gamma oscillatory activity in response to simulated brief pulses of ACh. ACh effects are simulated as transient modulation of the conductance of an M‐type K⁺current which is blocked by activation of muscarinic receptors and has significant effects on neuronal excitability. The ACh‐induced effects on the M current conductance,g_Ks, change network dynamics to promote the emergence of network gamma rhythmicity through a Pyramidal‐Interneuronal Network Gamma mechanism. Depending on connectivity strengths between and among E and I cells, gamma activity decays with the simulatedg_Kstransient modulation or is sustained in the network after theg_Kstransient has completely dissipated. We investigated the sensitivity of the emergent gamma activity to synaptic strengths, external noise and simulated levels ofg_Ksmodulation. To address recent experimental findings that cholinergic signaling is likely spatially focused and dynamic, we show that localizedg_Ksmodulation can induce transient changes of cellular excitability in local subnetworks, subsequently causing population‐specific gamma oscillations. These results highlight dynamical mechanisms underlying localization of ACh‐driven responses and suggest that spatially localized, cholinergically induced gamma may contribute to selectivity in the processing of competing external stimuli, as occurs in attentional tasks. 

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2. Dichotomy and Measures on Limit Sets of Anosov Groups

   https://doi.org/10.1093/imrn/rnad188  

   Lee, Minju; Oh, Hee (October 2023, International Mathematics Research Notices)

   Abstract Let $$G$$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $$\Gamma <G$$, we show that a $$\Gamma $$-conformal measure is supported on the limit set of $$\Gamma $$ if and only if its dimension is $$\Gamma $$-critical. This implies the uniqueness of a $$\Gamma $$-conformal measure for each critical dimension, answering the question posed in our earlier paper with Edwards [13]. We obtain this by proving a higher rank analogue of the Hopf–Tsuji–Sullivan dichotomy for the maximal diagonal action. Other applications include an analogue of the Ahlfors measure conjecture for Anosov subgroups. 

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3. Proof of the Goldberg–Seymour conjecture on edge–colorings of multigraphs

   https://doi.org/10.1007/s10878-025-01348-6  

   Chen, Guantao; Jing, Guangming; Zang, Wenan (September 2025, Journal of Combinatorial Optimization)

   Abstract Given a multigraph$$G=(V,E)$$, the edge-coloring problem (ECP) is to color the edges ofGwith the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer program, and its linear programming relaxation is referred to as the fractional edge-coloring problem (FECP). The optimal value of ECP (resp. FECP) is called the chromatic index (resp. fractional chromatic index) ofG, denoted by$$\chi '(G)$$(resp.$$\chi ^*(G)$$). Let$$\Delta (G)$$be the maximum degree ofGand let$$\Gamma (G)$$be the density ofG, defined by$$\begin{aligned} \Gamma (G)=\max \left\{ \frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hspace{5.69054pt}\textrm{and} \hspace{5.69054pt}\textrm{odd} \right\} , \end{aligned}$$whereE(U) is the set of all edges ofGwith both ends inU. Clearly,$$\max \{\Delta (G), \, \lceil \Gamma (G) \rceil \}$$is a lower bound for$$\chi '(G)$$. As shown by Seymour,$$\chi ^*(G)=\max \{\Delta (G), \, \Gamma (G)\}$$. In the early 1970s Goldberg and Seymour independently conjectured that$$\chi '(G) \le \max \{\Delta (G)+1, \, \lceil \Gamma (G) \rceil \}$$. Over the past five decades this conjecture, a cornerstone in modern edge-coloring, has been a subject of extensive research, and has stimulated an important body of work. In this paper we present a proof of this conjecture. Our result implies that, first, there are only two possible values for$$\chi '(G)$$, so an analogue to Vizing’s theorem on edge-colorings of simple graphs holds for multigraphs; second, although it isNP-hard in general to determine$$\chi '(G)$$, we can approximate it within one of its true value, and find it exactly in polynomial time when$$\Gamma (G)>\Delta (G)$$; third, every multigraphGsatisfies$$\chi '(G)-\chi ^*(G) \le 1$$, and thus FECP has a fascinating integer rounding property. 

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4. Large Monochromatic Components in Almost Complete Graphs and Bipartite Graphs

   https://doi.org/10.37236/9824  

   Füredi, Zoltán; Luo, Ruth (April 2021, The Electronic Journal of Combinatorics)

   Gyárfas proved that every coloring of the edges of $$K_n$$ with $t+1$ colors contains a monochromatic connected component of size at least $n/t$. Later, Gyárfás and Sárközy asked for which values of $$\gamma=\gamma(t)$$ does the following strengthening for almost complete graphs hold: if $$G$$ is an $$n$$-vertex graph with minimum degree at least $$(1-\gamma)n$$, then every $(t+1)$-edge coloring of $$G$$ contains a monochromatic component of size at least $n/t$. We show $$\gamma= 1/(6t^3)$$ suffices, improving a result of DeBiasio, Krueger, and Sárközy. 

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5. Independent dominating sets in graphs of girth five

   https://doi.org/10.1017/s0963548320000279  

   Harutyunyan, Ararat; Horn, Paul; Verstraete, Jacques (May 2021, Combinatorics, Probability and Computing)

   Abstract Let $$\gamma(G)$$ and $${\gamma _ \circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n -vertex graph of minimum degree at least d , then $$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$ In this paper the main result is that if G is any n -vertex d -regular graph of girth at least five, then $$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$ for some constant c independent of d . This result is sharp in the sense that as $$d \rightarrow \infty$$ , almost all d -regular n -vertex graphs G of girth at least five have $$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$ Furthermore, if G is a disjoint union of $${n}/{(2d)}$$ complete bipartite graphs $$K_{d,d}$$ , then $${\gamma_\circ}(G) = \frac{n}{2}$$ . We also prove that there are n -vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that $${\gamma_\circ}(G) \sim {n}/{2}$$ as $$d \rightarrow \infty$$ . Therefore both the girth and regularity conditions are required for the main result. 

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