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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. 

The fall armyworm, Spodoptera frugiperda, a globally invasive pest, demonstrates distinct immune adaptations across developmental stages and sexes, which are critical for its survival and adaptability. Using high-throughput RNA sequencing, this study systematically profiled 56 immune-related gene families, identifying 157 genes involved in Toll and Imd signaling pathways, and 185 genes associated with cellular immunity. Dynamic expression patterns were observed, with humoral immunity indices peaking during the third (L3) and fifth (L5) instars and diminishing in the pupal (P) and egg stages. In contrast, cellular immunity indices were highest in pupae and adult females, while the sixth instar (L6) and adult males exhibited the lowest immune capacity. Female adults displayed superior immune potential compared to males, reflecting evolutionary pressures tied to reproductive fitness. Notably, larval stages exhibited heightened immune gene expression, which aligns with their vulnerability to pathogens. Validation via qRT-PCR confirmed these transcriptomic trends, highlighting the modulation of immunity throughout development. These findings offer novel insights into the interplay between developmental progression and immune regulation in S. frugiperda. By elucidating these stage-specific immune responses, this study provides a robust framework for developing targeted pest management strategies aimed at mitigating the impact of this destructive pest. 

Harnessing instabilities of multicomponent multistable structural assemblies can potentially lead to scalable and reversible functionalities, which can be enhanced by exploring frustration. For instance, standard Kresling origami cells exhibit nontunable intrinsic energy landscapes determined by their geometry and material properties, limiting their adaptability after fabrication. To overcome this limitation, we introduce frustration to enable fine-tuning of the energy landscape and resulting deformation states. By prestressing the Kresling cell by means of special springs with individual control, we induce either global or localized (i.e., crease level) frustration, which allows changing the energy barrier (cell or assembly). We investigate the mechanical behavior of frustrated Kresling assemblies, both theoretically and experimentally, under various loading and boundary conditions. Our findings reveal that changing the frustration state leads to precise control of folding sequences, enabling previously inaccessible folding paths. The proposed concept paves the way for applications in mechanical metamaterials and other fields requiring highly programmable and reconfigurable systems – e.g., prosthetic limbs. 

We investigate the dynamics of a pair of rigid rotating helices in a viscous fluid, as a model for bacterial flagellar bundle and a prototype of microfluidic pumps. Combining experiments with hydrodynamic modelling, we examine how spacing and phase difference between the two helices affect their torque, flow field and fluid transport capacity at low Reynolds numbers. Hydrodynamic coupling reduces the torque when the helices rotate in phase at constant angular speed, but increases the torque when they rotate out of phase. We identify a critical phase difference, at which the hydrodynamic coupling vanishes despite the close spacing between the helices. A simple model, based on the flow characteristics and positioning of a single helix, is constructed, which quantitatively predicts the torque of the helical pair in both unbounded and confined systems. Finally, we show the influence of spacing and phase difference on the axial flux and the pump efficiency of the helices. Our findings shed light on the function of bacterial flagella and provide design principles for efficient low-Reynolds-number pumps. 

Degradable polymers are promising materials for use to reduce plastic waste and advance biomedical applications. However, to meet the demands of specific applications, tailoring the properties of degradable polymers through precise modification of their chemical structures is critical. Herein, we present a new class of degradable and functionalizable polyacetals synthesized by the ring-opening metathesis copolymerization (ROMP) of two commercially available monomers: dimethyl oxanorbornadiene-2,3-dicarboxylate (OND) and 4,7-dihydro-1,3-dioxepin (DXP). The resulting polyacetals are not only acid-degradable but also readily and efficiently functionalizable via thia–Michael addition, yielding degradable polymer materials with various functional groups and tunable thermal properties.