  In this paper, we consider third order Lovelock gravity with a cosmological
constant term in an n-dimensional spacetime $\mathcal{M}^{4}\times
\mathcal{K}^{n-4}$, where $\mathcal{K}^{n-4} $ is a constant curvature space.
We decompose the equations of motion to four and higher dimensional ones and
find wormhole solutions by considering a vacuum $\mathcal{K}^{n-4} $ space.
Applying the latter constraint, we determine the second and third order
Lovelock coefficients and the cosmological constant in terms of specific
parameters of the model, such as the size of the extra dimensions. Using the
obtained Lovelock coefficients and $\Lambda$, we obtain the 4-dimensional
matter distribution threading the wormhole. Furthermore, by considering the
zero tidal force case and a specific equation of state, given by $\rho =(\gamma
p-\tau )/[\omega (1+\gamma )]$, we find the exact solution for the shape
function which represents both asymptotically flat and non-flat wormhole
solutions. We show explicitly that these wormhole solutions in addition to
traversibility satisfy the energy conditions for suitable choices of parameters
and that the existence of a limited spherically symmetric traversable wormhole
with normal matter in a 4-dimensional spacetime, implies a negative effective
cosmological constant.
