16-vertex graphs with automorphism groups A4 and A5 from the icosahedron

Abstract

The article deals with the problem of finding vertex-minimal graphs with a given automorphism group. We exhibit two undirected 16-vertex graphs having automorphism groups A4 and A5. It improves Babai's bound for A4 and the graphical regular representation bound for A5. The graphs are constructed using projectivisation of the vertex-face graph of the icosahedron.

Keywords

• graph, icosahedron, hemi-icosahedron, automorphism group, alternating group