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Einstein-Gauss-Bonnet traversable wormholes satisfying the weak energy condition

In this paper, we explore higher-dimensional asymptotically flat wormhole geometries in the framework of Gauss-Bonnet (GB) gravity and investigate the effects of the GB term, by considering a specific radial-dependent redshift function and by imposing a particular equation of state. This work is motivated by previous assumptions that wormhole solutions were not possible for the $k=1$ and $α< 0$ case, where $k$ is the sectional curvature of an $(n-2)$-dimensional maximally symmetric space, and $α$ is the Gauss-Bonnet coupling constant. However, we emphasize that this discussion is purely based on a nontrivial assumption that is only valid at the wormhole throat, and cannot be extended to the entire radial-coordinate range. In this work, we provide a counterexample to this claim, and find for the first time specific solutions that satisfy the weak energy condition throughout the entire spacetime, for $k=1$ and $α< 0$. In addition to this, we also present other wormhole solutions which alleviate the violation of the WEC in the vicinity of the wormhole throat.