$16$-vertex graphs with automorphism groups $A_{4}$ and $A_{5}$ from icosahedron

TL;DR

This paper constructs two 16-vertex graphs with automorphism groups A4 and A5, improving known bounds by leveraging the icosahedron's structure.

Contribution

It presents explicit 16-vertex graphs with automorphism groups A4 and A5, surpassing previous bounds and using projectivisation of the icosahedron's vertex-face graph.

Findings

Graphs have automorphism groups A4 and A5

Improves Babai's bound for A4

Enhances graphical regular representation bound for A5

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 $A_{4}$ and $A_{5}$. It improves the Babai's bound for $A_{4}$ and the graphical regular representation bound for $A_{5}$. The graphs are constructed using projectivisation of the vertex-face graph of icosahedron.

Full text

1 Introduction

1.1 Outline

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 $\mu(G)$ the minimal number of vertices of undirected graphs having automorphism group isomorphic to $G$, $\mu(G)=\min\limits_{\Gamma:Aut(\Gamma)\simeq G}|V(\Gamma)|$. It is known [1] that $\mu(G)\leq 2|G|$, for any finite group $G$ which is not cyclic of order $3,4$ or $5$. See Babai [2] and Cameron [4], for expositions of this area. There are groups which admit a graphical regular representation, for such groups $\mu(G)\leq|G|$. For some recent work see [6], [7], [9].

For alternating groups $A_{n}$ $\mu(A_{n})$ is known for $n\geq 13$, see Liebeck [10]. If $n\equiv 0\ or\ 1(mod\ 4)$, then $\mu(A_{n})=2^{n}-n-2$. Additionally, for $n\geq 5$ $A_{n}$ admits a graphical regular representation, see [13]. Thus for $A_{5}$ the best published estimate until now seemed to be $\mu(A_{5})\leq 60$.

In this paper we exhibit graphs $\Gamma_{i}=(V,E_{i})$, $i\in\{4,5\}$, such that $|V|=16$ and $Aut(\Gamma_{i})\simeq A_{i}$. The graph $\Gamma_{5}$ (also denoted $\Pi_{I}$) is listed in [5] together with order of its automorphism group. These $\mu$ values are less than the Babai’s bound for groups $A_{4}$ and $A_{5}$. For $A_{5}$ our graph has fewer vertices than the graphical regular representation. The new graphs are based on projectivisation of vertex-face incidence relation of icosahedron.

1.2 Notations

We use standard notations for undirected graphs, see Diestel [8]. A bipartite graph $\Gamma$ with vertex partition sets $V_{1}$ and $V_{2}$ is denoted as $\Gamma=(V_{1},V_{2},E)$.

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 $\mathbb{R}^{3}$ not containing the origin, then its image under a projectivisation map to $P(\mathbb{R}^{3})$ is denoted by $\pi(S)$ or $[S]$, $[S]=\bigcup_{x\in S}[x]$.

2 Main results

2.1 Vertex-face graphs of polyhedra

Definition 1**.**

Let $P=(V,E,F)$ be a polyhedron. An undirected bipartite graph $\Gamma_{P}=(V,F,I)$ is the vertex-face graph of $P$ if $v\sim f$ iff $v\in V$, $f\in F$ and $v\in f$. In other words, $\Gamma_{P}$ corresponds to the vertex-face incidence relation in $V\times F$.

Definition 2**.**

Let $S=(V,E,F)$ be a centrally symmetric polyhedron. Let $S$ be positioned in $\mathbb{R}^{3}$ so that its center is at $(0,0,0)$. We call the undirected bipartite graph $\Pi_{S}=([V],[F],I_{p})$ projective vertex-face graph if for any $v_{p}\in[V]$, $f_{p}\in[F]$ we have $v_{p}\sim f_{p}$ iff $v\in f$ for some $v\in\pi^{-1}(v_{p})$ and $f\in\pi^{-1}(f_{p})$.

2.2 Projective vertex-face graph of icosahedron and $A_{5}$

Let $I=(V,E,F)$ be a regular icosahedron. Denote by $Rot(I)\leq SO(3)$ the group of rotational symmetries of $I$ - rotations of $\mathbb{R}^{3}$ preserving $V$ and $E$. It is known that $Rot(I)\simeq A_{5}$. $\Pi_{I}$ is shown in Fig.1. We note that $\Pi_{I}$ can be interpreted as the vertex-face graph of hemi-icosahedron, see [12].

Fig.1. - $\Pi_{I}$.

Proposition 3**.**

Let $I$ be regular icosahedron. Then $Aut(\Pi_{I})\simeq A_{5}$.

Proof.

The stated fact can be checked using an appropriate software, such as Magma, see [3]. Nevertheless we give a proof based on the geometric construction. We prove that $Rot(I)\simeq Aut(\Pi_{I})$ in two steps.

First we prove that there is a subgroup in $Aut(\Pi_{I})$ isomorphic to $Rot(I)$. We show that there is an injective group morphism $f:Rot(I)\stackrel{{\scriptstyle f_{1}}}{{\rightarrow}}Aut(\Gamma_{I})\stackrel{{\scriptstyle f_{2}}}{{\rightarrow}}Aut(\Pi_{I}).$ $f_{1}:Rot(I)\rightarrow Aut(\Gamma_{I})$ maps every $\rho\in Rot(I)$ to $f_{1}(\rho)\in Aut(\Gamma_{I})$ which is the permutation of $V\cup F$ induced by $\rho$: $f_{1}(\rho)(x)=\rho(x)$ for any $x\in V\cup F$. Rotations of $I$ preserve the vertex-face incidence relation and $f_{1}$ is a group morphism. $f_{2}:Aut(\Gamma_{I})\rightarrow Aut(\Pi_{I})$ maps every $\varphi\in Aut(\Gamma_{I})$ to $\varphi_{P}\in Aut(\Pi_{I})$ defined by the rule $\varphi_{P}([x])=[\varphi(x)]$ for any $x\in V(\Gamma_{I})$. 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 show that $|Aut(\Pi_{I})|\leq 60$ by a counting argument. Every vertex $v\in[V]$ is contained in a subgraph $\sigma(v)$ shown in Fig.2.

Fig.2. - $\sigma(v)$.

All $\Pi_{I}$-vertices in $[V]$ have degree $5$, all $\Pi_{I}$-vertices in $[F]$ have degree $3$. It follows that $[V]$ and $[F]$ both are unions of $Aut(\Pi_{I})$-orbits. $v$ can be mapped by a $\Pi_{I}$-automorphism in at most $6$ possible ways. After fixing the image of $v$ it follows again by $Aut(\Pi_{I})$-invariance of $[V]$ that the subgraph $\sigma(v)$ can be mapped in at most $10$ ways. Any permutation of $[V]$ by an automorphism determines a unique permutation of $[F]$. Thus $|Aut(\Pi_{I})|\leq 60$. We have shown that $Aut(\Pi_{I})=f(Rot(I))\simeq A_{5}$. ∎

Remark 4**.**

A graph isomorphic to $\Pi_{I}$ is listed without discussion of automorphism group in [5] as one of connected edge-transitive bipartite graphs, ET16.5.

2.3 A modification of the projective vertex-face graph of icosahedron and $A_{4}$

Since $A_{5}$ has subgroups isomorphic to $A_{4}$, we can try to modify $\Pi_{I}$ so that the automorphism group of the modified graph is isomorphic to $A_{4}$. We find generators for a subgroup $H\leq Rot(I)$, such that $H\simeq A_{4}$, and add $3$ extra edges to $\Pi_{I}$ which are permuted only by elements of $H$.

Denote by $I_{1}$ the polyhedral ($1$-skeleton) graph of $I$, $Aut(I_{1})\simeq Sym(I)\simeq A_{5}\times\mathbb{Z}_{2}$. An isomorphism $Sym(I)\rightarrow Aut(I_{1})$ takes a symmetry $f$ to $f|_{I_{1}}$.

Proposition 5**.**

Choose a $6$-subset of vertices $W=\{O,A,B,C,D,E\}\subseteq V(I)$ such that $I_{1}[W]$ is isomorphic to the $5$-wheel, see Fig.3.

Fig.3. - $I_{1}[W]$.

Define an undirected graph $\Xi_{I}=([V]\cup[F],I_{p}\cup J)$ by adding $3$ edges to $\Pi_{I}$: $J=\{[A]\sim[C],[B]\sim[O],[D]\sim[E]\}$, see Fig.4. Then $Aut(\Xi_{I})\simeq A_{4}$.

Fig.4. - extra edges.

Proof.

Consider the subgroup $H=\langle r_{1},r_{2}\rangle\leq Rot(I)$ generated by two rotations: $r_{1}$ - rotation by $\frac{2\pi}{3}$ radians around the line passing through the center of the face $OCD$ and the center of $I$, $r_{2}$ - rotation by $\pi$ radians around the line passing through the center of the edge $OB$ and the center of $I$.

It can be checked that $H\simeq A_{4}$. Note that the vertices $O,A,B,C,D,E$ in Fig.3 represent the $6$ projective classes of $V$.

We have to show that $Aut(\Xi_{I})\simeq H$. First we show that $H\leq Aut(\Xi_{I})$. $\Xi_{I}$ differs from $\Pi_{I}$ by $3$ extra edges. It suffices to note by direct inspection that $r_{1}$ permutes these extra edges and $r_{2}$ fixes each of them. To show that $Aut(\Xi_{I})\leq H$ we observe that any additional rotation $r^{\prime}$ does not permute these three new edges and thus $r^{\prime}\not\in Aut(\Xi_{I})$. ∎

Remark 6**.**

If $D$ is dodecahedron then $\Pi_{D}\simeq\Pi_{I}\simeq A_{5}$.

Acknowledgements

We used $Magma$, see Bosma et al. [3], and $Nauty$, available at $http://cs.anu.edu.au/~{}bdm/data/$, see McKay and Piperno [11]. The author thanks Valentina Beinarovica for her assistance.