16-vertex graphs with automorphism groupsA4 and A5 from the icosahedron

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.

This article addresses a problem in graph representation theory of finite groups -finding undirected graphs with a given automorphism group and minimal number of vertices. Denote by µ(G) the minimal number of vertices of undirected graphs having automorphism group isomorphic to G, µ(G) = min Γ:Aut(Γ) G |V (Γ)|. It is known [1] that µ(G) ≤ 2|G|, for any finite group G which is not cyclic of order 3, 4 or 5. See Babai [2] for an exposition of this area. There are groups which admit a graphical regular representation, for such groups µ(G) ≤ |G|. For some recent work see [4]. For alternating groups A n µ(A n ) is known for n ≥ 13, see Liebeck [6]. If n ≡ 0 or 1(mod 4), then µ(A n ) = 2 n − n − 2. Additionally, for n ≥ 5 A n admits a graphical regular representation, see [8]. Thus for A 5 the best published estimate until now seemed to be µ(A 5 ) ≤ 60.
In this paper we exhibit graphs Γ i = (V, E i ), i ∈ {4, 5}, such that |V | = 16 and Aut(Γ i ) A i . Γ 4 (also denoted Ξ I ) improves Babai's bound for A 4 . Γ 5 (also denoted Π I ) has fewer vertices than the graphical regular representation of A 5 . Γ 5 is listed in [3] together with the order of its automorphism group. The new graphs are based on projectivisation of the vertex-face incidence relation of the regular icosahedron. We use standard notation for undirected graphs, see Diestel [5]. A bipartite graph Γ with vertex partition sets V 1 and V 2 is denoted as Γ = (V 1 , V 2 , E). Given a polyhedron P , we denote its vertex, edge and face sets as V = V (P ), E = E(P ) and F = F (P ), respectively. We can think of P as the triple (V, E, F ). If S is a subset of R 3 not containing the origin, then its image under the projectivisation map to P (R 3 ) is denoted by π(S) or

Main results
In this section we define objects used for our construction -projective vertex-face graphs. We prove that the automorphism group of the projective vertex-face graph of the regular icosahedron is A 5 . We further show that after adding three extra edges we get a graph with the automorphism group A 4 .

Vertex-face graphs of polyhedra
and v ∈ f . In other words, Γ P corresponds to the vertex-face incidence relation in V × F . Definition 1.2. Let S = (V, E, F ) be a centrally symmetric polyhedron. Let S be positioned in R 3 so that its center is at (0, 0, 0). We call the undirected bipartite graph

Projective vertex-face graph of the icosahedron and A 5
Let I = (V, E, F ) be the regular icosahedron. Define Γ 5 = Π I , it is shown in Fig.1, an adjacency matrix of Π I is given in Appendix A. Π I can be interpreted in terms of the hemi-icosahedron, see [7]. Proof. We prove that Rot(I) Aut(Π I ) in two steps. First we show that there is a subgroup in Aut(Π I ) isomorphic to Rot(I) -the group of rotational symmetries of I, rotations of R 3 preserving V and E. It is known that Rot(I) A 5 . There is an injective group morphism f : Rot(I) for any x ∈ V ∪ F . Rotations of I preserve the vertex-face incidence relation and f 1 is a group morphism. f 2 : Aut(Γ I ) → Aut(Π I ) maps every ϕ ∈ Aut(Γ I ) to ϕ P ∈ Aut(Π I ) defined by the rule ϕ P ([x]) = [ϕ(x)] for any x ∈ V (Γ I ). Projectivization and composition commute therefore f 2 is a group morphism. f is injective since there is no nontrivial rotation of I sending each vertex to another vertex in the same projective class.
In the second step we prove that |Aut(Π I )| ≤ 60 by a counting argument. Every vertex v ∈ [V ] is contained in a subgraph σ(v) shown in Fig.2. Remark 1.1. A graph isomorphic to Π I is listed without discussion of its construction and automorphism group in [3] as ET16.5.

A modification of the projective vertex-face graph of the icosahedron and A 4
Since A 5 has subgroups isomorphic to A 4 , we can try to modify Π I so that the automorphism group of the modified graph is isomorphic to A 4 . We find generators for a subgroup H ≤ Rot(I), such that H A 4 , and add three extra edges to Π I which are permuted only by elements of H.

Appendices
A -An adjacency matrix of Π I