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An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graphGat least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. But what if Euler had to take a bus? In a temporal graph$$\varvec{(G,\lambda )}$$$(G,\lambda )$, with$$\varvec{\lambda : E(G)}\varvec{\rightarrow } \varvec{2}^{\varvec{[\tau ]}}$$$\lambda :E\left(G\right)\to 2^{\left[\tau \right]}$, an edge$$\varvec{e}\varvec{\in } \varvec{E(G)}$$$e\in E\left(G\right)$is available only at the times specified by$$\varvec{\lambda (e)}\varvec{\subseteq } \varvec{[\tau ]}$$$\lambda \left(e\right)\subseteq \left[\tau \right]$, in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether$$\varvec{(G,\lambda )}$$$(G,\lambda )$has a temporal walk or trail is polynomial, while deciding whether it has a local trail is$$\varvec{\texttt {NP}}$$$\mathrm{NP}$-complete even if$$\varvec{\tau = 2}$$$\tau =2$. In contrast, in the general case, solving any of these problems is$$\varvec{\texttt {NP}}$$$\mathrm{NP}$-complete, even under very strict hypotheses. We finally give$$\varvec{\texttt {XP}}$$$\mathrm{XP}$algorithms parametrized by$$\varvec{\tau }$$$\tau$for walks, and by$$\varvec{\tau +tw(G)}$$$\tau +tw\left(G\right)$for trails and local trails, where$$\varvec{tw(G)}$$$tw\left(G\right)$refers to the treewidth of$$\varvec{G}$$$G$.

 

Using inelastic X-ray scattering beyond the dipole limit and hard X-ray photoelectron spectroscopy we establish the dual nature of the U 5 f electrons in U M 2 S i 2 (M = Pd, Ni, Ru, Fe), regardless of their degree of delocalization. We have observed that the compounds have in common a local atomic-like state that is well described by the U 5 f 2 configuration with the Γ 1 ( 1 ) and Γ 2 quasi-doublet symmetry. The amount of the U 5 f 3 configuration, however, varies considerably across the U M 2 S i 2 series, indicating an increase of U 5f itineracy in going from M = Pd to Ni to Ru and to the Fe compound. The identified electronic states explain the formation of the very large ordered magnetic moments in U P d 2 S i 2 and U N i 2 S i 2 , the availability of orbital degrees of freedom needed for the hidden order in U R u 2 S i 2 to occur, as well as the appearance of Pauli paramagnetism in U F e 2 S i 2 . A unified and systematic picture of the U M 2 S i 2 compounds may now be drawn, thereby providing suggestions for additional experiments to induce hidden order and/or superconductivity in U compounds with the tetragonal body-centered T h C r 2 S i 2 structure.