Charles Delorme wrote:

Besides the K4*X8, there is a beautiful graph, with a high symmetry;
alas, I do not remember its author (perhaps Wegner?).
Take the pentagonal dodecahedron, (degree 3, 20 vertices), add an edge
between each pair of opposite vertices, add 10 new vertices that correspond
to regular tetrahedra inscribed in the dodecahedron. Connect them
to the corresponding vertices. Part these 10 tetrahedra into two sets
of 5, each covering the 20 vertices of the dodecahedron. Connect opposite
tetrahedra. Add two
adjacent vertices and connect each of them to the tetrahedra in one part of
the partition.

0..19: the vertices of the dodecahedron
20..29: the tetrahedra
30,31: the two last vertices.

Here is the incidence list

0:1,2,3,18,20,29
1:0,4,5,17,23,27
2:0,6,7,19,21,28
3:0,8,9,16,22,26
4:1,9,10,13,21,25
5:1,6,11,14,24,26
6:2,5,12,15,22,25
7:2,8,10,13,24,27
8:3,7,11,14,23,25
9:3,4,12,15,24,28
10:4,7,11,19,22,29
11:5,8,10,16,20,28
12:6,9,13,16,23,29
13:4,7,12,17,20,26
14:5,8,15,17,21,29
15:6,9,14,19,20,27
16:3,11,12,18,21,27
17:1,13,14,18,22,28
18:0,16,17,19,24,25
19:2,10,15,18,23,26
20:0,11,13,15,25,30
21:2,4,14,16,26,30
22:3,6,10,17,27,30
23:1,8,12,19,28,30
24:5,7,9,18,29,30
25:4,6,8,18,20,31
26:3,5,13,19,21,31
27:1,7,15,16,22,31
28:2,9,11,17,23,31
29:0,10,12,14,24,31
30:20,21,22,23,24,31
31:25,26,27,28,29,30