  An almost complex torus manifold is a $2n$-dimensional compact connected
almost complex manifold equipped with an effective action of a real
$n$-dimensional torus $T^n \simeq (S^1)^n$ that has fixed points. For an almost
complex torus manifold, there is a labeled directed graph which contains
information on weights at the fixed points and isotropy spheres.
  Let $M$ be a 6-dimensional almost complex torus manifold with Euler number 6.
We show that two types of graphs occur for $M$, and for each type of graph we
construct such a manifold $M$, proving the existence. Using the graphs, we
determine the Chern numbers and the Hirzebruch $\chi_y$-genus of $M$.
