An Icosian Hamiltonian path
This vZome model is an approximation of the one featured in this instructional YouTube video. These yellow and green balls correspond roughly to the two layers of balls forming a border on either side of the Hamiltonian path in the video. The actual Hamiltonian path is shown here by the 12 struts or equivalently, the perimiter of the folded net of triangular panels shown inside the balls. The panels constitute exactly half of the faces and all of the vertices of the icosahedron.
See also Icosian game on wikipedia