Abstract
We prove that, if \(\varGamma \) is a finite connected 3valent vertextransitive, or 4valent vertex and edgetransitive graph, then either \(\varGamma \) is part of a wellunderstood family of graphs, or every nonidentity automorphism of \(\varGamma \) fixes at most 1/3 of the edges. This answers a question proposed by Primož Potočnik and the third author.
1 Introduction
Potočnik and Spiga have proved in [11] that, if \(\varGamma \) is a finite connected 3valent vertextransitive graph, or a 4valent vertex and edgetransitive graph then, unless \(\varGamma \) belongs to a wellknown family of graphs, every nonidentity automorphism of \(\varGamma \) fixes at most 1/3 of the vertices. In the same work, they have proposed a similar investigation with respect to the edges of the graph, see [11, Problem 1.7]. In this paper we solve this problem.
Theorem 1
Let \(\varGamma \) be a finite connected 4valent vertex and edgetransitive graph admitting a nonidentity automorphism fixing more than 1/3 of the edges. Then one of the following holds:

1.
\(\varGamma \) is isomorphic to the complete graph on 5 vertices;

2.
\(\varGamma \) is isomorphic to a Praeger–Xu graph C(r, s), for some r and s with \(3s < 2r3\).
Theorem 2
Let \(\varGamma \) be a finite connected 3valent vertextransitive graph admitting a nonidentity automorphism fixing more than 1/3 of the edges. Then \(\varGamma \) is isomorphic to a split Praeger–Xu graph SC(r, s), for some r and s with \(3s < 2r2\).
We refer to Sect. 2.3 for the definition of the ubiquitous Praeger–Xu graphs and for their splitting. The bound in Theorem 2 is sharp. For instance, each 3valent graph admitting a nonidentity automorphism fixing elementwise a complete matching has the aforementioned property. For valency 4, we conjecture that the bound 1/3 in Theorem 1 can be strengthened to 1/4, by eventually including some more small exceptional graphs in part (1).
Theorems 1 and 2 rely on the following grouptheoretic fact:
Theorem 3
[10, Theorem 1.1] Let G be a finite transitive permutation group on \(\varOmega \) containing no nonidentity normal subgroup of order a power of 2. Suppose there exists \(\omega \in \varOmega \) such that the stabilizer \(G_\omega \) of \(\omega \) in G is a 2group. Then, every nonidentity element of G fixes at most 1/3 of the points.
The main results of this paper and the results in [11] show that, besides small exceptions or wellunderstood families of graphs, nonidentity automorphisms of 3valent or 4valent vertextransitive graphs cannot fix many vertices or edges, where “too many” in this context has to be considered as a linear function on the number of vertices (and, even then, with a small caveat for 4valent graphs, because of the assumption of edgetransitivity). In our opinion, the difficulty in having a unifying theory of vertextransitive graphs of small valency admitting nonidentity automorphisms fixing too many vertices or edges is due to our lack of understanding possible generalizations of Praeger–Xu graphs, that is, a family of vertextransitive graphs of bounded valency playing the role of Praeger–Xu graphs. It seems to us that this is a recurrent problem in the theory of groups acting on finite graphs of bounded valency. A general investigation in this direction, but with much weaker bounds and only for arctransitive graphs, is in [7].
Investigations on the number of fixed points of graph automorphisms do have interesting applications. For instance, very recently Potočnik, Toledo and Verret [14] pivoting on the results in [11] have proved remarkable results on the cycle structure of general automorphisms of 3valent vertextransitive and 4valent arctransitive graphs.
1.1 Structure of the paper
In Sect. 2, we introduce some basic terminology and, in particular, we introduce the Praeger–Xu graphs and their splitting. Then, we start in Sect. 3 with some preliminary results. In Sect. 4, we prove Theorem 1 and, in Sect. 5, we prove Theorem 2.
2 The players
2.1 Basic grouptheoretic notions
Given a permutation g on a set \(\varOmega \), we write \(\mathrm {Fix}(\varOmega ,g)\) for the set of fixed points of g , i.e.
and we write \(\mathrm {fpr} (\varOmega ,g)\) for the fixedpoint ratio of g , i.e.
A permutation group G on \(\varOmega \) is said to be semiregular if the identity is the only element fixing some point. When G is semiregular and transitive on \(\varOmega \), the group G is regular on \(\varOmega \).
Given a permutation group G of \(\varOmega \) and a partition \(\Sigma \) of \(\varOmega \), we say that \(\Sigma \) is Ginvariant if \(\sigma ^g\in \pi \), for every \(\sigma \in \Sigma \). Given a normal subgroup N of G, the orbits of N on \(\varOmega \) form a Ginvariant partition, which we denote by \(\varOmega / N\).
We present here a useful lemma involving the notion just defined.
Lemma 1
[11, Lemma 1.17] Let G be a group acting transitively on \(\varOmega \) and let \(\Sigma \) be a Ginvariant partition of \(\varOmega \). For \(g\in G\), let \(g^\Sigma \) be the permutation of \(\Sigma \) induced by g. Then \( \mathrm {fpr}(\varOmega , g) \le \mathrm {fpr}(\Sigma , g^\Sigma ).\) In particular, if \(N\unlhd G\), then \(\mathrm {fpr}(\varOmega , g) \le \mathrm {fpr}(\varOmega /N, Ng)\).
2.2 Basic graphtheoretic notions
In this paper, a digraph is a binary relation
where \(A\varGamma \subseteq V\varGamma \times V\varGamma \). We refer to the elements of \(V\varGamma \) as vertices and to the elements of \(A\varGamma \) as arcs. A graph is a finite simple undirected graph, i.e. a pair
where \(V\varGamma \) is a finite set of vertices, and \(E\varGamma \) is a set of unordered pairs of \(V\varGamma \), called edges. In particular, a graph can be thought of as a digraph where the binary relation is symmetric and contains no loops. Given a nonnegative integer s, an sarc of \(\varGamma \) is an ordered set of \(s+1\) adjacent vertices with any three consecutive elements pairwise distinct. When \(s=0\), an sarc is simply a vertex of \(\varGamma \); when \(s=1\), an sarc is simply an arc, that is, an oriented edge.
The girth of \(\varGamma \), denoted by \(g(\varGamma )\), is the minimum length of a cycle in \(\varGamma \).
We denote by \(\varGamma (v)\) the neighbourhood of the vertex v. The size of \(\varGamma (v)\) is the valency of v. We are mainly dealing with regular graphs, that is, with graphs where \(\varGamma (v)\) is constant as v runs through the elements of \(V\varGamma \). In these cases, we refer to the valency of the graph.
Let \(\varGamma \) be a graph, let G be a subgroup of the automorphism group \(\mathrm {Aut}(\varGamma )\) of \(\varGamma \), let \(v \in V\varGamma \) and let \(w\in \varGamma (v)\). We denote by \(G_v\) the stabilizer of the vertex v, by \(G_{\{v,w\}}\) the setwise stabilizer of the edge \(\{v,w\}\), by \(G_{vw}\) the pointwise stabilizer of the edge \(\{v,w\}\) (that is, the stabilizer of the arc (v, w) underlying the edge \(\{v,w\}\)). The group \(G_v\) acts on \(\varGamma (v)\) and we denote by \(G_v^{[1]}\) the kernel of the action of \(G_v\) on \(\varGamma (v)\). Now, the permutation group induced by \(G_v\) on \(\varGamma (v)\) is denoted by \(G_v^{\varGamma (v)}\) and we have
When G acts transitively on the set of sarcs of \(\varGamma \), we say that G is sarctransitive. When \(s=0\), we say that G is vertextransitive and, when \(s=1\), we say that G is arctransitive. Moreover, when G acts regularly on the set of sarcs of \(\varGamma \) we emphasize this fact by saying that G is sarcregular.
When G acts transitively on \(E\varGamma \), we say that G is edgetransitive. Finally, when G is edge and vertextransitive, but not arctransitive, we say that G is halfarctransitive. This name comes from the fact that G has two orbits on ordered pairs of adjacent vertices of \(\varGamma \) (a.k.a. arcs), each orbit containing precisely one of the two arcs underlying each edge.
We say that \(\varGamma \) is vertex, edge or arctransitive when \(\mathrm {Aut}(\varGamma )\) is vertex, edge or arctransitive.
Let G be a finite group and let S be a subset of G. The Cayley digraph on G with connection set S is the digraph \(\varGamma :=\mathrm {Cay}(G,S)\) having vertex set G and where \((g,h)\in A\varGamma \) if and only if \(gh^{1}\in G\). Now, \(\mathrm {Cay}(G,S)\) is a symmetric binary relation if and only if S is inverse closed, that is, \(S=S^{1}\) where \(S^{1}:=\{s^{1}\mid s \in S\}\). Observe that the right regular representation of G acts as a group of automorphisms on \(\mathrm {Cay}(G,S)\).
2.3 Praeger–Xu graphs
In this and in the next section, we introduce the infinite families of graphs appearing in our main theorems. We introduce the 4valent PraegerXu graphs C(r, s) through their directed counterpart defined in [16]. Further details on Praeger–Xu graphs can be found in [2, 4, 17]. We also advertise [5], where the authors have begun a thorough investigation of Praeger–Xu graphs, motivated by the recurrent appearance of these objects in the theory of groups acting on graphs.
Let \(r\ge 3 \) be an integer. Then \(\mathbf {C}(r,1)\) is the lexicographic product of a directed cycle of length r with the edgeless graph on 2 vertices. In other words, \(V\mathbf {C}(r,1)={\mathbb {Z}}_r\times {\mathbb {Z}}_2\), and the two arcs starting in (x, i) end in \((x+1,0)\) and in \((x+1,1)\). For any \(2\le s\le r1\), \(V\mathbf {C}(r,s)\) is defined as the set of all \((s1)\)arcs of \(\mathbf {C}(r,1)\), and \((v_0,v_1,\dots ,v_{s1})\in V\mathbf {C}(r,s)\) is the beginning point of the two arcs ending in \((v_1, v_2,\dots ,v_{s1},u)\) and in \((v_1, v_2,\dots ,v_{s1},u')\), where u and \(u'\) are the two vertices of \(\mathbf {C}(r,1)\) that prolong the \((s1)\)arc \((v_1, v_2,\dots ,v_{s1})\). The Praeger–Xu graph C(r, s) is then defined as the nonoriented underlying graph of \(\mathbf {C}(r,s)\). It can be verified that C(r, s) is a connected 4valent graph with \(r2^s\) vertices and \(r2^{s+1}\) edges.
We describe the automorphisms of C(r, s). Some automorphism of C(r, s) arises from the action of \(\mathrm {Aut}(C(r,1))\) on the set of sarcs of C(r, 1). Let \(i\in {\mathbb {Z}}_r\) and let \(\tau _i\) be the transposition on \(V\mathbf {C}(r,1)\) swapping the vertices (i, 0) and (i, 1) and fixing the remaining vertices. Since \(\tau _i\) is an automorphism of \(\mathbf {C}(r,1)\), it is immediate to extended the action of \(\tau _i\) to C(r, 1) and to C(r, s). We define the group
and throughout this paper the symbol K will always refer to this group for some C(r, s). Focusing on the cyclic nature of the PraegerXu graphs, it is also natural to define on \(V\mathbf {C}(r,1)\) the permutations \(\rho \) and \(\sigma \) as follows
While \(\rho \) is an automorphism of \(\mathbf {C}(r,1)\), \(\sigma \) is an automorphism of C(r, 1) but not of \(V\mathbf {C}(r,1)\). Moreover, observe that the group \(\langle \rho ,\sigma \rangle \) normalizes K. Define
and, as for K, the symbols H and \(H^+\) will always refer to these groups. Clearly \(H\cong C_2 \wr D_r\) is a group of automorphisms of C(r, s) and \(H^+\cong C_2 \wr C_r\) is a group of automorphisms of \(\mathbf {C}(r,s)\). Moreover, H acts vertex and edgetransitively on C(r, s) (and so does \(H^+\) on \(\mathbf {C}(r,s)\)), but not 2arctransitively.
Lemma 2
Using the notation above, \(\mathrm {Aut}( \mathbf {C}(r,s)) = H^+\) and, if \(r\ne 4\), \(\mathrm {Aut}(C(r,s))=H\). Moreover,
Proof
It follows from [16, Theorem 2.8] and [17, Theorem 2.13] when \(p=2\). \(\square \)
The Praeger–Xu graphs also admit the following algebraic characterization.
Lemma 3
Let \(\varGamma \) be a finite connected 4valent graph and let G be a vertex and edgetransitive group of automorphisms of \(\varGamma \). If G has an abelian normal subgroup which is not semiregular on \(V\varGamma \), then \(\varGamma \) is isomorphic to a Praeger–Xu graph C(r, s), for some integers r and s.
Proof
It follows by [16, Theorem 2.9] and [17, Theorem 1] upon setting \(p=2\). \(\square \)
2.4 Split Praeger–Xu graphs
For our purposes, the split Praeger–Xu graphs are obtained from the Praeger–Xu graphs via the splitting operation which was introduced in [12, Construction 9], and which we will comment upon in Sect. 5.
Here we give an explicit description of SC(r, s). Split any vertex of \(\mathbf {C}(r,s)\) into two copies, say \(v_+\) and \(v_\). For any arc of \(\mathbf {C}(r,s)\) of the form (v, u), let \(v_+\) be adjacent to \(v_\) and \(u_\). From the complementary perspective, the neighbourhood of \(v_\) is made up of \(v_+\) plus the two vertices \(w_+\) such that (w, v) is an arc of \(\mathbf {C}(r,s)\).
3 Preliminary results
3.1 Graphtheoretical considerations
In this section, we develop our tool box that extends outside the scope of proving our main theorems.
Lemma 4
Let \(\varGamma \) be a connected kvalent graph, with \(k\ge 3\), and let G be an sarctransitive group of automorphisms of \(\varGamma \). Then \(2s\le g(\varGamma ) +2\). In particular, the girth of \(\varGamma \) is greater than s.
Proof
The first part of the statement is [1, Proposition 17.2]. The second one is an immediate computation if \(s\ge 2\), and it follows from \(g(\varGamma )\ge 3\) if \(s=1\). \(\square \)
Lemma 5
Let \(\varGamma \) be a finite connected graph and let \(v\in V\varGamma \) be a vertex. For each \(w\in \varGamma (v)\), suppose there exists \(t_w\) automorphism of \(\varGamma \) such that \(v^{t_w}=w\). Then \(T:=\langle t_w\mid w\in \varGamma (v)\rangle \) is vertextransitive on \(\varGamma \).
Proof
Let \(u\in V\varGamma \). As \(\varGamma \) is connected, we prove the existence of \(t_u\in T\) with \(v^{t_u}=u\) arguing by induction on the minimal distance \(d:=d(v,u)\) from v to u in \(\varGamma \). When \(d=0\), that is, \(v=u\), we may take \(t_u\) to be the identity of T. Suppose then \(d>0\). Let \(v_0,\ldots ,v_d\) be a path of distance d from \(v=v_0\) to \(u=v_d\) in \(\varGamma .\) Now, \(d(v,v_{d1})=d1\) and hence, by induction, there exists \(t\in T\) with \(v^{t}=v_{d1}\). Set \(u':=u^{t^{1}}\). As \(u=v_d\in \varGamma (v_{d1})\), we have
By hypothesis, \(t_{u'}\in T\) and \(v^{t_{u'}}=u'\). Therefore, \(v^{t_{u'}t}=u'^{t}=u\) and we may take \(t_u:=t_{u'}t\). \(\square \)
Lemma 6
[3, Lemma 3.3.3] Let \(\varGamma \) be a finite connected vertextransitive graph of valency k. Then \(\varGamma \) is kedgeconnected, i.e. \(\varGamma \) remains connected upon eliminating any m edges, with \(m\le k1\).
A general result on the fixedpoint ratio of Cayley graphs can be proven regardless of the valency.
Lemma 7
Let G be a finite group, let S be an inverse closed nonempty subset of G, let \(\varGamma :=\mathrm {Cay}(G,S)\) and let \(g\in G\setminus \{1\}\). If \(\mathrm {fpr}(E\varGamma ,g)\ne 0\), then \(g^2=1\) and
where \(g^G:=\{hgh^{1}\mid h\in G\}\) is the conjugacy class of g in G. In particular, \(\mathrm {fpr}(E\varGamma ,g)\le 1/S\) and the equality is attained if and only if \(g^G\subseteq S\).
Proof
Suppose \(\mathrm {fpr}(E\varGamma ,g)\ne 0\). We let \(\mathbf{C}_G(g)\) denote the centralizer of g in G.
For each \(s\in S\), let \(E_s:=\{\{x,sx\}\mid x\in G\}\). Observe that \(E_s\) is a complete matching of \(\varGamma \) and that \(\{E_s\mid s\in S\}\) is a partition of the edge set \(E\varGamma \).
Let \(s\in S\). Suppose \(E_s\cap \mathrm {Fix}(E\varGamma ,g)\ne \emptyset \) and fix \(\{{\bar{x}},s{\bar{x}}\}\in E_s\cap \mathrm {Fix}(E\varGamma ,g)\). As g fixes the edge \(\{{\bar{x}},s{\bar{x}}\}\), we have \({\bar{x}}g=s{\bar{x}}\) and \(s{\bar{x}}g={\bar{x}}\). We deduce \(g^2=1\) and \(s={\bar{x}}g{\bar{x}}^{1}\). In other words, g has order 2 and g has a conjugate in S. Now, for every \(\{x,sx\}\in E_s\), with a similar computation, we obtain that \(\{x,sx\}\in \mathrm {Fix}(E\varGamma ,g)\) if and only if \(s=xgx^{1}\). Thus \({\bar{x}}g{\bar{x}}^{1}=xgx^{1}\) and \(x\in {\bar{x}}\mathbf{C}_G(g)\). In particular, \(E_s\cap \mathrm {Fix}(E\varGamma ,g)=\{\{{\bar{x}}h,s{\bar{x}}h\}\mid h\in \mathbf{C}_G(x)\}\) and hence
The previous paragraph has established that g has order 2. Moreover, for each \(s\in S\), \(E_s\cap \mathrm {Fix}(E\varGamma ,g)\ne \emptyset \) if and only if \(s\in g^G\). Furthermore, in the case that \(s\in g^G\), the cardinality of \(E_s\cap \mathrm {Fix}(E\varGamma ,g)\) does not depend on s and equals \(\mathbf{C}_G(g)/2\). Therefore,
Since \(g^G\cap S\le g^G\), we have \(\mathrm {fpr}(E\varGamma ,S)\le 1/S\). Moreover, the equality is attained if and only if \(g^G\cap S=g^G\), that is, \(g^G\subseteq S\). \(\square \)
The next lemma studies the nature of fixed edges in a Praeger–Xu graph.
Lemma 8
Let \(\varGamma =C(r,s)\) be a Praeger–Xu graph and let \(g\in \mathrm {Aut}(\varGamma )\) with \(g\ne 1\) and with \(\mathrm {fpr}(E\varGamma , g)> 1/3\). Then \(3s < 2r3\) and, either \(g\in K\) or \((r,s)=(4,1)\). In particular, g fixes an edge if and only if g fixes both of its ends. (The group K is defined in Sect. 2.3.)
Proof
The lexicographic product \(C(4,1)\cong K_{4,4}\) admits automorphisms h fixing 8 edges and hence \(\mathrm {fpr}(E\varGamma ,h)=8/16=1/2>1/3\). (The nonidentity elements h in \(\mathrm {Aut}(C(4,1))\) with \(\mathrm {fpr}(E\varGamma ,h)>1/3\) are not necessarily in K, but they fix an edge if and only if they fix both of its ends.) Similarly, it can be verified that, for every \(h\in \mathrm {Aut}(C(4,2))\) with \(h\ne 1\), we have \(\mathrm {fpr}(E\varGamma ,h)\le 8/32=1/4\). Furthermore, for every \(h\in \mathrm {Aut}(C(4,3))\) with \(h\ne 1\), we have \(\mathrm {fpr}(E\varGamma , h)=8/64=1/8\). In particular, when \(r=4\), the result follows from these computations.
Suppose \(r\ne 4\). By Lemma 2, \(\mathrm {Aut}(\varGamma )=H=K\langle \rho , \sigma \rangle \). In particular,
Denote by \(\varDelta _x\) the set of \((s1)\)arcs in \(\mathbf {C}(r,1)\) starting at (x, 0) or at (x, 1). From the definition of the vertex set of C(r, s), we have \(\varDelta _x\subseteq VC(r,s)\), \(\varDelta _x=2^s\) and
We claim that the subgraph induced by \(\varGamma \) on \(\varDelta _x\cup \varDelta _{x+1}\) is the disjoint union of cycles of length 4. In fact, consider the \((s1)\)arcs in \(\varDelta _x\) parameterized as
for some \(y_i \in {\mathbb {Z}}_2\). In \(\varGamma \), they are both adjacent to
Since the induced subgraph is 2valent, these elements form a cycle of length 4, which is a connected component of the induced graph. Moreover, \(\varDelta _x\) is a Korbit, and, for any \(x\in {\mathbb {Z}}_r\),
We start by proving that \(g\in K\).
Suppose \(\varepsilon =0\). Let \(\{a,b\}\in \mathrm {Fix}(E\varGamma ,g)\). Replacing a with b if necessary, we may suppose that \(a\in \varDelta _x\) and \(b\in \varDelta _{x+1}\), for some \(x\in {\mathbb {Z}}_r\). If \(a^g=a\) and \(b^g=b\), we have \(\varDelta _x^g=\varDelta _{x}\) and \(\varDelta _{x+1}^g=\varDelta _{x+1}\). Now, (3.1) yields \(x+i=x\) and \((x+1)+i=x+1\), that is, \(i=0\). Therefore \(g\in K\). Similarly, if \(a^g=b\) and \(b^g=a\), we have \(\varDelta _x^g=\varDelta _{x+1}\) and \(\varDelta _{x+1}^g=\varDelta _{x}\). Now, (3.1) yields \(x+i=x+1\) and \((x+1)+i=x\), that is, \(2=0\). However, this implies \(r=2\), which is a contradiction because \(r\ge 3\).
Suppose \(\varepsilon =1\). Since \(\langle \rho ,\sigma \rangle \) is a dihedral group of order 2r, replacing g by a suitable conjugate if necessary, we may suppose that either r is odd and \(i=0\), or r is even and \(i\in \{0,1\}\).
Assume \(i=0\). Let \(\{a,b\}\in \mathrm {Fix}(E\varGamma ,g)\). As above, replacing a with b if necessary, we may suppose that \(a\in \varDelta _x\) and \(b\in \varDelta _{x+1}\), for some \(x\in {\mathbb {Z}}_r\). If \(a^g=a\) and \(b^g=b\), we have \(\varDelta _x^g=\varDelta _{x}\) and \(\varDelta _{x+1}^g=\varDelta _{x+1}\). Now, (3.1) yields \(xs+1=x\) and \((x+1)s+1=x+1\), that is, \(2=0\). However, this gives rise to the contradiction \(r=2\). Similarly, if \(a^g=b\) and \(b^g=a\), we have \(\varDelta _x^g=\varDelta _{x+1}\) and \(\varDelta _{x+1}^g=\varDelta _{x}\). Now, (3.1) yields \(xs+1=x+1\) and \((x+1)s+1=x\), that is, \(2x+s=0\). When r is odd, the equation \(2x+s=0\) has only one solution in \({\mathbb {Z}}_r\) and, when r is even, the equation \(2x+s=0\) has either zero or two solutions in \({\mathbb {Z}}_r\) depending on whether s is odd or even. Recalling that the subgraph induced by \(\varGamma \) on \(\varDelta _x\cup \varDelta _{x+1}\) is a disjoint union of cycles of length 4, and noticing that g fixes at most 2 edges of any cycle, we obtain that
In both cases, we have \(\mathrm {fpr}(E\varGamma ,g)\le 1/4\), which is a contradiction.
Assume \(i=1\). Observe that this implies that r is even. Here the analysis is entirely similar. Let \(\{a,b\}\in \mathrm {Fix}(E\varGamma ,g)\). As above, replacing a with b if necessary, we may suppose that \(a\in \varDelta _x\) and \(b\in \varDelta _{x+1}\), for some \(x\in {\mathbb {Z}}_r\). If \(a^g=a\) and \(b^g=b\), we have \(\varDelta _x^g=\varDelta _{x}\) and \(\varDelta _{x+1}^g=\varDelta _{x+1}\). Now, (3.1) yields \((x+1)s+1=x\) and \((x+2)s+1=x\), that is, \(2=0\). However, this gives rise to the usual contradiction \(r=2\). Similarly, if \(a^g=b\) and \(b^g=a\), we have \(\varDelta _x^g=\varDelta _{x+1}\) and \(\varDelta _{x+1}^g=\varDelta _{x}\). Now, (3.1) yields \((x+1)s+1=x+1\) and \((x+2)s+1=x\), that is, \(2x+s+1=0\). As r is even, the equation \(2x+s+1\) has either zero or two solutions in \({\mathbb {Z}}_r\) depending on whether s is even or odd. Recalling that the subgraph induced by \(\varGamma \) on \(\varDelta _x\cup \varDelta _{x+1}\) is a disjoint union of cycles of length 4, and noticing that g fixes at most 2 edges of any cycle, we obtain that
Thus, we have \(\mathrm {fpr}(E\varGamma ,g)\le 1/4\), which is a contradiction.
Since \(g\in K\), if g fixes the edge \(\{a,b\}\in E\varGamma \), then g fixes both endvertices a and b. It remains to show that \(3s<2r3\). Notice that \(\tau _i\) moves precisely those \((s1)\)arcs of \(\mathbf {C}(r,1)\) that pass through one of the vertices (i, 0) or (i, 1). Therefore, \(\tau _i\), as an automorphism of C(r, s), fixes all but \(s2^s\) vertices, thus it fixes all but those \((s+1)2^{s+1}\) edges which are incident with such vertices. Since any element in K is obtained as a product of some \(\tau _i\), such an element fixes at most as many edges as a single \(\tau _i\). Hence
\(\square \)
Lemma 9
Let \(\varGamma =C(r,s)\) be a Praeger–Xu graph, let G be a vertex and edgetransitive group of automorphism of \(\varGamma \) containing a nonidentity element g fixing more than 1/3 of the edges and with G not 2arctransitive. Then G is \(\mathrm {Aut}(\varGamma )\)conjugate to a subgroup of H as defined in Sect. 2.3.
Proof
By Lemma 8, \(3s < 2r3\). If \(r\ne 4\), then by Lemma 2 we have \(G\le \mathrm {Aut}(\varGamma )=H\). When \(r=4\), then inequality \(3s<2r3\) implies \(s=1\). Now, the veracity of this lemma can be verified with a computation in \(\mathrm {Aut}(C(4,1))=\mathrm {Aut}(K_{4,4})=S_4\wr S_2\). \(\square \)
Lemma 10
[11, Lemma 1.14] Let \(\varGamma \) be a finite connected 4valent graph, let G be a vertex and edgetransitive group of automorphisms of \(\varGamma \), and let N be a minimal normal subgroup of G. If N is a 2group and \(\varGamma /N\) is a cycle of length at least 3, then \(\varGamma \) is isomorphic to a Praeger–Xu graph C(r, s) for some integers r and s.
4 Proof of Theorem 1
In this section we prove Theorem 1. Our proof is divided into two cases, depending on whether \(\varGamma \) admits a group of automorphisms acting 2arctransitively or not.
4.1 Proof of Theorem 1 when \(\varGamma \) is 2arctransitive
The following lemma involves four graphs not yet considered in this paper, so it is worth to spend some ink here to describe them.

The complete graph \(K_5\) is the only sporadic example arising in Theorem 1, its automorphism group is \(S_5\) and each transposition in \(S_5\) fixes 4 edges out of 10.

The graph \(K_{5,5}5K_2\) is obtained deleting a complete matching from the complete bipartite graph \(K_{5,5}\), its automorphism group is \(S_5\times C_2\) and every nonidentity automorphism fixes at most 6 edges out of 20.

The hypercube \(Q_4\) is the Cayley graph
$$\begin{aligned} Q_4 := \mathrm {Cay} ({\mathbb {Z}}_2 ^4, \{(1,0,0,0),(0,1,0,0), (0,0,1,0), (0,0,0,1)\}). \end{aligned}$$A nonidentity automorphism of \(Q_4\) fixes at most 8 edges out of 32.

The graph BCH is the bipartite complement of the Heawood graph. The vertices of BCH can be identified with the 7 points and the 7 lines of the Fano plane. The incidence in the graph is given by the antiflags in the plane, i.e. the point p is adjacent to the line L if, and only if, \(p\notin L\). The automorphism group of BCH is isomorphic to \(\mathrm {SL}_3(2).2\). A nonidentity automorphism of BCH fixes at most 4 edges out of 28.
Lemma 11
Let \(\varGamma \) be a finite connected 4valent 2arctransitive graph of girth at most 4, i.e. \(g(\varGamma )\in \{3,4\}\). Then one of the following holds:

1.
\(g(\varGamma )=3\) and \(\varGamma \) is isomorphic to the complete graph \(K_5\);

2.
\(g(\varGamma )=4\) and \(\varGamma \) is isomorphic to \(K_{4,4}\cong C(4,1)\);

3.
\(g(\varGamma )=4\) and \(\varGamma \) is isomorphic to \(K_{5,5}5K_2\), \(Q_4\) or BCH.
Proof
Let v be a vertex, let \(\varGamma (v)=\{w_1,w_2,w_3,w_4\}\) be its neighbourhood and let \(G:=\mathrm {Aut}(\varGamma )\).
First, assume \(g(\varGamma )=3\). Without loss of generality, suppose \(w_1\) and \(w_2\) are adjacent. Since G is 2arctransitive, \(G_v\) is 2transitive on \(\varGamma (v)\). Hence \(w_i\) is adjacent to \(w_j\) for any \(i\ne j\). Thus \(\varGamma \cong K_5\) and part (1) holds.
Now, suppose \(g(\varGamma )=4\). We need to recall the classification arising from [15, Theorem 3.3]. If \(\varDelta \) is a 4valent edgetransitive graph, then one of the following holds

(i)
each vertex in \(\varDelta \) is contained in exactly one 4cycle,

(ii)
there exist two distinct vertices \(v_1,v_2\) with \(\varDelta (v_1)=\varDelta (v_2)\),

(iii)
\(\varDelta \) is isomorphic to \(K_{5,5}5K_2\), \(Q_4\) or BCH.
We consider these three possibilities for \(\varGamma \) in turn. Up to a permutation of the indices, there exists \(u\in \varGamma (w_1)\cap \varGamma (w_2)\) such that \((v,w_1,u,w_2)\) is a 4cycle. Since \(G_v^{\varGamma (v)}\) is 2transitive, there exists \(g\in G_v\) with \((w_1,w_2)^g=(w_3,w_4)\). Therefore, \((v,w_1,u,w_2)^g=(v,w_3,u^g,w_4)\) is a 4cycle different from \((v,w_1,u,w_2)\). Thus part (i) is excluded. If \(\varGamma \) satisfies (ii), then [15, Lemma 4.3] gives that \(\varGamma \) is isomorphic to C(r, 1) for some integer r. From Lemma 2, C(r, 1) is 2arctransitive only when \(r=4\); therefore we obtain part (2). If \(\varGamma \) satisfies part (iii), then we obtain the examples in part (3). \(\square \)
Definition 1
Let \(\varGamma \) be a finite connected 4valent graph and let g be an automorphism of \(\varGamma \). We partition \(E\varGamma \) with respect to the action of g.

We let \(A(\varGamma ,g)\) be the set of edges which are pointwise fixed by g, that is, \(\{a,b\}\in A(\varGamma ,g)\) if and only if \(\{a,b\}\in E\varGamma \), \(a^g=a\) and \(b^g=b\);

we let \(F(\varGamma ,g):=\mathrm {Fix}(E\varGamma ,g)\setminus A(\varGamma ,g)\), that is, \(\{a,b\}\in F(\varGamma ,g)\) if and only if \(\{a,b\}\in E\varGamma \), \(a^g=b\) and \(b^g=a\);

we let \(N(\varGamma ,g):=E\varGamma \setminus \mathrm {Fix}(E\varGamma ,g)\).
We let \(\varGamma [g]\) denote the subgraph of \(\varGamma \) induced by \(\varGamma \) on the vertices which are incident with edges in \(A(\varGamma ,g)\). The edgeset of \(\varGamma [g]\) is \(A(\varGamma ,g)\) and its vertices are 1, 2 or 4valent. Given \(i\in \{1,2,4\}\), we let \(V_i(\varGamma ,g)\) denote the set of vertices of \(\varGamma [g]\) having valency i.
Lemma 12
Let \(\varGamma \) be a finite connected 4valent graph of girth \(g(\varGamma )\ge 5\) and let g be an automorphism of \(\varGamma \). Then \(2F(\varGamma ,g)+4V_1(\varGamma ,g)+3V_2(\varGamma ,g)+V_4(\varGamma ,g)\le V\varGamma \).
Proof
We let
Since \(V_1(\varGamma ,g),V_2(\varGamma ,g),V_4(\varGamma ,g),{\mathcal {F}},{\mathcal {N}}\) are pairwise disjoint and since \({\mathcal {F}}=2F(\varGamma ,g)\), it suffices to show that \({\mathcal {N}}\ge 3V_1(\varGamma ,g)+2V_2(\varGamma ,g)\).
We construct an auxiliary graph \(\varDelta \). The vertex set of \(\varDelta \) is \(V_1(\varGamma ,g)\cup V_2(\varGamma ,g)\cup {\mathcal {N}}\) and we declare a vertex \(v\in V_1(\varGamma ,g)\cup V_2(\varGamma ,g)\) adjacent to a vertex \(u\in {\mathcal {N}}\) if \(\{v,u\}\in E\varGamma \). By construction, \(\varDelta \) is bipartite with parts \(V_1(\varGamma ,g)\cup V_2(\varGamma ,g)\) and \({\mathcal {N}}\).
Given \(v\in V_1(\varGamma ,g)\), the automorphism g acts as a 3cycle on \(\varGamma (v)\). Let \(v_1,v_2,v_3\in \varGamma (v)\) forming the 3cycle of g. Then \(\{v,v_1\},\{v,v_2\},\{v,v_3\}\in N(\varGamma ,g)\) and hence \(v_1,v_2,v_3\in {\mathcal {N}}\). This shows that each vertex in \(V_1(\varGamma ,g)\) has three neighbours in \({\mathcal {N}}\). Similarly, each vertex in \(V_2(\varGamma ,g)\) has two neighbours in \({\mathcal {N}}\). As \(g(\varGamma )>4\), we have \(g(\varDelta )>4\) and hence \(3V_1(\varGamma ,g)+2V_2(\varGamma ,g)\le {\mathcal {N}}\), because \(\varDelta (v)\cap \varDelta (v')=\emptyset \) for any two distinct vertices \(v,v'\in V_1(\varGamma ,g)\cup V_2(\varGamma ,g)\). \(\square \)
Let B, L and R be groups, and let \(\iota _L: B \rightarrow L\) and \(\iota _R: B\rightarrow R\) be injective homomorphisms of groups. The pair \((\iota _L,\iota _R)\) is said to be an amalgam. When B is a subgroup of both L and R, we can think of \(\iota _L\) and \(\iota _R\) as the inclusion mappings. In this case, the amalgam is determined by the triple (L, B, R) and, in this paper, this is the point of view we take.
Let (L, B, R) be an amalgam, we say that its index is the couple
Moreover, (L, B, R) is said to be faithful if no subgroup of B is normal in L and in R. When the index is precisely (k, 2), for some positive integer k, (L, B, R) is said to be 2transitive if the action of L on the right cosets of B by right multiplication is 2transitive.
Observe that, if \(\varGamma \) is a finite connected Garctransitive graph of valency k, then for any \(v\in V\varGamma \) and \(w\in \varGamma (v)\), the triplet
is a faithful amalgam of index (k, 2).
Finite faithful 2transitive amalgams of index (4, 2) have been studied in detail by Potočnik in [9]. We use this work to deduce some properties on \(\mathrm {Fix}(E\varGamma ,g)\).
Lemma 13
Let \(\varGamma \) be a finite connected 4valent graph, let G be an sarctransitive group of automorphisms of \(\varGamma \) with \(s\ge 2\) and let \(g\in G\) fixing pointwise the sarc \((v_0,\ldots ,v_{s1})\). If G is not \((s+1)\)arctransitive and g fixes pointwise \(\varGamma (v_0)\cup \varGamma (v_{s1})\), then \(g=1\).
Proof
If G is sarcregular, then \(g=1\) because g fixes an sarc. Using [9], we see that there are 6 amalgams such that G is not sarcregular. For each of these remaining amalgams a casebycase computation shows that the only automorphism leaving the neighbourhood of each end of a given sarc fixed is the identity map. \(\square \)
Lemma 14
Let \(\varGamma \) be a finite connected 4valent graph of girth \(g(\varGamma )\ge 5\), let G be a 2arctransitive group of automorphisms of \(\varGamma \) such that \(G_v^{[1]}\cap G_w^{[1]}\) is a 3group, for any two distinct vertices at distance at most 2, and let \(g\in G\setminus \{1\}\). Then \(3V_4(\varGamma ,g)\le 3V_1(\varGamma ,g)+V_2(\varGamma ,g)\).
Proof
Assume that the vertices in \(V_4(\varGamma ,g)\) are at pairwise distance more than 2. Then any two such vertices share no common neighbour. In particular, \(\bigcup _{v\in V_4(\varGamma ,g)} \varGamma (v)\) has cardinality \(4V_4(\varGamma ,g)\) and is contained in \(V_1(\varGamma ,g)\cup V_2(\varGamma ,g)\). Therefore, \(4V_4(\varGamma ,g)\le V_1(\varGamma ,g)+V_2(\varGamma ,g)\) and the lemma immediately follows in this case.
Assume that there exist two distinct vertices v and w of \(V_4(\varGamma ,g)\) having distance at most 2. In particular, \(g\in G_v^{[1]}\cap G_w^{[1]}\) and hence g has order a power of 3, because \(G_v^{[1]}\cap G_w^{[1]}\) is a 3group. Observe that \(V_2(\varGamma ,g)=\emptyset \) because an element of order 3 in a local group cannot fix exactly two elements. Let \(s\ge 2\) such that G is sarctransitive, but not \((s+1)\)arctransitive.
Suppose \(\varGamma [g]\) is not a forest. Then \(\varGamma [g]\) contains an \(\ell \)cycle C. As \(V_2(\varGamma ,g)=\emptyset \), the vertices of C are elements of \(V_4(\varGamma ,g)\). From Lemma 4, we have \(g(\varGamma [g])\ge g(\varGamma )\ge s+1\) and hence, from C, we can extract an sarc whose ends lie in \(V_4(\varGamma ,g)\), contradicting Lemma 13.
Suppose \(\varGamma [g]\) is a forest. Let c be the number of connected components of \(\varGamma [g]\). From Euler’s formula, we have \(V\varGamma [g]E\varGamma [g]=c\). Clearly, \(V\varGamma [g]=V_1(\varGamma ,g)+V_4(\varGamma ,g)\). Let \({\mathcal {S}}:=\{(v,w)\in V\varGamma [g]\times V\varGamma [g]\mid \{v,w\}\in E\varGamma [g]\}\). Then
It follows that \(2V_4(\varGamma ,g)=V_1(\varGamma ,g)2c<V_1(\varGamma ,g)\). \(\square \)
Proof
(Proof of Theorem 1 when \(\varGamma \) is 2arctransitive) Let \(\varGamma \) be a finite connected 4valent 2arctransitive graph admitting a nonidentity automorphism g with \(\mathrm {fpr}(E\varGamma ,g)>1/3\) and let \(G:=\mathrm {Aut}(\varGamma )\).
If \(g(\varGamma )\le 4\), then the proof follows from Lemma 11 and from the remarks at the beginning of Sect. 4.1. Therefore, for the rest of the proof we suppose that \(g(\varGamma )>4\). Since \(4V\varGamma =2E\varGamma \), we have
where in the last inequality we have used Lemma 12.
We claim that, for any two distinct vertices \(v,w\in V\varGamma \) at distance at most 2 one of the following holds

(i)
\(G_v^{[1]}\cap G_w^{[1]}\) is a 3group;

(ii)
the pair \((\varGamma ,G)\) defines the amalgam
$$\begin{aligned} \left( S_3\times S_4, S_3\times S_3, (S_3\times S_3)\rtimes C_2 \right) , \end{aligned}$$moreover, if \(d(v,w)=1\), then \(G_v^{[1]}\cap G_w^{[1]}=1\) and, if \(d(v,w)=2\), then \(G_v^{[1]}\cap G_w^{[1]}\) is isomorphic to \(C_2\).
The claim follows with a casebycase computation on the finite faithful 2transitive amalgams of index (4, 2) classified in [9]. We now divide the proof according to (i) and (ii).
Suppose that (i) holds. From Lemma 14, we have \(3V_4(\varGamma ,g)\le 3V_1(\varGamma ,g)+V_2(\varGamma ,g)\). Using this inequality and (4.1), we obtain \(\mathrm {fpr}(E\varGamma ,g)\le 1/4<1/3\), which is a contradiction.
Suppose that (ii) holds. If there exist two distinct vertices v and w in \(V_4(\varGamma ,g)\) with \(d(v,w)=1\), then \(g\in G_v^{[1]}\cap G_w^{[1]}=1\), which is a contradiction. Assume there exist two distinct vertices v and w in \(V_4(\varGamma ,g)\) with \(d(v,w)=2\). Then \(g\in G_v^{[1]}\cap G_w^{[1]}\cong C_2\) and hence g has order 2. This implies \(V_1(\varGamma ,g)=\emptyset \) because an involution in a local group cannot fix only one element. Since the subgraph induced by \(\varGamma [g]\) on \(V_4(\varGamma ,g)\) has no edges and since each vertex in \(V_4(\varGamma ,g)\) has valency 4, we deduce \(4V_4(\varGamma ,g)\le E\varGamma [g]=V_2(\varGamma ,g)+2V_4(\varGamma ,g)\). Using this inequality and (4.1), we obtain \(\mathrm {fpr}(E\varGamma ,g)<1/3\), which is a contradiction.
Finally, assume that the vertices in \(V_4(\varGamma ,g)\) are at pairwise distance more than 2. Then any two such vertices share no common neighbour. In particular, \(\bigcup _{v\in V_4(\varGamma ,g)} \varGamma (v)\) has cardinality \(4V_4(\varGamma ,g)\) and is contained in \(V_1(\varGamma ,g)\cup V_2(\varGamma ,g)\). Therefore, \(4V_4(\varGamma ,g)\le V_1(\varGamma ,g)+V_2(\varGamma ,g)\). Using this inequality and (4.1), we obtain \(\mathrm {fpr}(E\varGamma ,g)\le 1/4<1/3\), which is a contradiction. \(\square \)
4.2 Proof of Theorem 1 when \(\varGamma \) is not 2arctransitive
To conclude the proof of Theorem 1, we argue by induction on \(V\varGamma \).
Let \(\varGamma \) be a finite connected vertex and edgetransitive 4valent graph admitting a nonidentity automorphism g fixing more than 1/3 of the edges and with \(G:=\mathrm {Aut}(\varGamma )\) not 2arctransitive. If \(\varGamma \) is isomorphic to a Praeger–Xu graph, then part (2) of Theorem 1 holds. Therefore, for the rest of the argument, we suppose that \(\varGamma \) is not isomorphic to C(r, s), for any choice of r and s with \(r\ge 3\) and \(1\le s\le r1\).
Let \(v\in V\varGamma \). Since G is not 2arctransitive, \(G_v^{\varGamma (v)}\) is not 2transitive on \(\varGamma (v)\). Since G is vertex and edgetransitive, we obtain that either \(G_v^{\varGamma (v)}\) is transitive or \(G_v^{\varGamma (v)}\) has two orbits of cardinality 2. In both cases, we deduce that \(G_v^{\varGamma (v)}\) is a 2group. As \(\varGamma \) is connected, it follows that \(G_v\) is a 2group.
If G has no nonidentity normal subgroups having cardinality a power of 2, Theorem 3 (applied to the faithful and transitive action of G on \(E\varGamma \)) contradicts \(\mathrm {fpr}(E\varGamma ,g)>1/3\). Thus, G has a minimal normal 2subgroup N.
As \(\varGamma \) is not isomorphic to a Praeger–Xu graph, Lemma 3 yields that N acts semiregularly on \(V\varGamma \). Consider the quotient graph \(\varGamma /N\) and observe that, as G is vertex and edgetransitive, \(\varGamma /N\) has valency 0, 1, 2 or 4.
If \(\varGamma /N\) has valency 0, then N is transitive on \(V\varGamma \). Thus N is vertexregular on \(\varGamma \). As \(\varGamma \) is connected of valency 4, N is generated by at most 4 elements and hence \(V\varGamma =N\) divides \(2^4\). If \(\varGamma /N\) has valency 1, then N has two orbits on \(V\varGamma \). Moreover, [11, Lemma 1.15] implies that \(V\varGamma =2N\) divides 128. In both cases, the statement can be checked computationally by inspecting the candidate graphs from the census of all 4valent vertex and edgetransitive graphs of small order, see [12, 13]. If \(\varGamma /N\) has valency 2, then we contradict Lemma 10. Therefore, for the rest of the proof, we may suppose that \(\varGamma /N\) has valency 4.
Let K be the kernel of the action of G on \(V\varGamma /N\). Since the quotient graph is not degenerate, \(K_v=1\). Thus \(K=K_vN=N\). In particular, G/N acts faithfully as a group of automorphisms on \(\varGamma /N\). Moreover, G/N acts vertex and edgetransitively on \(\varGamma /N\), but not 2arctransitively. Observe that \(g\notin N\), because the elements in N fix no edge of \(\varGamma \). Thus Ng is not the identity automorphism of \(\varGamma /N\) and, by Lemma 1, we have \(\mathrm {fpr}(E\varGamma /N,Ng)>1/3\). Our inductive hypothesis on \(V\varGamma \) implies that \(\varGamma /N\) is isomorphic to \(K_5\) or to a Praeger–Xu graph C(r, s) with \(3s < 2r3\).
Assume \(\varGamma /N\cong K_5\). Now, \(\mathrm {Aut}(K_5)=S_5\) and \(S_5\) contains a unique conjugacy class of subgroups which are vertex and edgetransitive, but not 2transitive (namely, the Frobenius groups of order 20). Therefore, G/N is isomorphic to a Frobenius group of order 20. In particular, as N is an irreducible module for a Frobenius group of order 20, we get \(N\le 16\). We deduce \(V\varGamma \le 10\cdot 16=160\) and, as above, the statement can be checked computationally by inspecting the census of all 4valent vertex and edgetransitive graphs of small order.
Assume \(\varGamma /N\cong C(r,s)\), for some r and s with \(3s<2r3\). From Lemma 9, G/N is \(\mathrm {Aut}(\varGamma /N)\)conjugate to a subgroup of H as defined in Sect. 2.3. Without loss of generality, we can identify G/N with such a subgroup, so that \(G/N\le H\). Now, we first deal with the exceptional case \((r,s)=(4,1)\). As G/N is a 2group and N is a minimal normal subgroup of G, we deduce \(N=2\) and hence \(V\varGamma =V\varGamma /NN=8\cdot 2=16\). Now, the proof follows inspecting the vertex and edgetransitive graphs of order 16. Therefore, for the rest of the argument, we suppose \((r,s)\ne (4,1)\). Now, Lemma 8 implies \(Ng\in K\le H^+\). Denote by X the group \(G/N\cap H^+\). This group is a halfarctransitive group of automorphisms of \(\varGamma /N\) and, since \(H:H^+=2\), we have \(G/N:X\le 2\). Denote by \(G^+\) the preimage of X with respect to the quotient projection \(G\rightarrow G/N\), so that \(G^+/N\cong X\). Now, \(G^+\) acts halfarctransitively on \(\varGamma \) and, from \(Ng\in X\), we see that \(g\in G^+\). In particular, replacing G with \(G^+\) if necessary, in the rest of our argument we may suppose that \(G=G^+\), that is, \(G/N\le H^+\).
By Lemma 8, all the edges fixed in \(\varGamma /N\) by Ng are fixed as arcs. Therefore, all the edges fixed in \(\varGamma \) by g are fixed as arcs.
Considering the graph induced by \(\varGamma \) on \(\mathrm {Fix}(V\varGamma ,g)\), we deduce \( 2\mathrm {Fix}(E\varGamma ,g) \le 4\mathrm {Fix}(V\varGamma ,g)\). In particular, if \(\mathrm {Fix}(V\varGamma ,g)\le V\varGamma /3\), then
which is a contradiction. Therefore \(\mathrm {fpr}(V\varGamma , g)>1/3\). Now, the hypothesis of Lemma 2.3 in [11] are satisfied. Therefore, [11, Lemma 2.3] implies that \(\varGamma \) is a Praeger–Xu graph, which is our final contradiction.
5 Proof of Theorem 2
We now turn our attention to finite connected 3valent vertextransitive graphs. We divide the proof of Theorem 2 in three cases, which we now describe. Let \(\varGamma \) be a finite connected 3valent vertextransitive graph, let \(G:=\mathrm {Aut}(\varGamma )\) and let \(v\in V\varGamma \). The local group \(G_v^{\varGamma (v)}\) is a subgroup of the symmetric group of degree 3 and we divide the proof of Theorem 2 depending on the structure of \(G_v^{\varGamma (v)}\). When \(G_v^{\varGamma (v)}=1\), the connectivity of \(\varGamma \) implies \(G_v=1\) and hence G acts regularly on \(V\varGamma \). In this case an observation of Sabidussi [18] yields that \(\varGamma \) is Cayley graph over G. We deal with this case in Sect. 5.1. When \(G_v^{\varGamma (v)}\) is cyclic of order 2, [12] has established a fundamental relation between \(\varGamma \) and a certain finite connected 4valent graph; in Sect. 5.2, we exploit this relation and Theorem 1 to deal with this case. When \(G_v^{\varGamma (v)}\) is transitive, \(\varGamma \) is arctransitive and we use the amazing result of Tutte concerning the structure of \(G_v\) to deal with this case in Sect. 5.3.
5.1 Proof of Theorem 2 when the local group is the identity
Let \(\varGamma \) be a finite connected 3valent vertextransitive graph, let \(v\in V\varGamma \), let \(G:=\mathrm {Aut}(\varGamma )\) and let \(g\in G\setminus \{1\}\). Assume that \(G_v^{\varGamma (v)}=1\). Lemma 7 yields \(\mathrm {fpr}(E\varGamma ,g)\le 1/3\) and hence Theorem 2 holds in this case.
5.2 Proof of Theorem 2 when the local group is cyclic of order 2
In our proof of this case, we need to refer to two families of 3valent Cayley graphs. Given \(n\in {\mathbb {N}}\) with \(n\ge 3\), the prism \(\mathrm {Pr}_n\) is the Cayley graph
Similarly, given \(n\in {\mathbb {N}}\) with \(n\ge 2\), the Möbius ladder \(\mathrm {Mb}_n\) is the Cayley graph
For these two classes of graphs the proof of Theorem 2 follows with a computation. When \(n\ne 4\), the automorphism group of \(\mathrm {Pr}_n\) is isomorphic to \(D_n\times C_2\) and, for each \(x\in \mathrm {Aut}(\mathrm {Pr}_n)\) with \(x\ne 1\), it can be verified that \(\mathrm {fpr}(E\mathrm {Pr}_n,x)\le 1/3\), see also Lemma 7. The case \(n=4\) is exceptional, because \(\mathrm {Pr}_4\cong Q_4\) is 2arctransitive and hence \(\mathrm {Pr}_4\) is of no concern to us here. Similarly, when \(n\notin \{2,3\}\), the automorphism group of \(\mathrm {Mb}_n\) is isomorphic to \(D_{2n}\) and, for each \(x\in \mathrm {Aut}(\mathrm {Mb}_n)\) with \(x\ne 1\), it can be verified that \(\mathrm {fpr}(E\mathrm {Mb},x)\le 1/3\), again see also Lemma 7. The cases \(n\in \{2,3\}\) are exceptional, because \(\mathrm {Mb}_2\cong K_4\) and \(\mathrm {Mb}_3\) are 2arctransitive and hence are of no concern to us here.
Now, let \(\varGamma \) be a finite connected 3valent vertextransitive graph not isomorphic to \(\mathrm {Pr}_n\) and not isomorphic to \(\mathrm {Mb}_n\), let \(v\in V\varGamma \), let \(G:=\mathrm {Aut}(\varGamma )\) and let \(g\in G\setminus \{1\}\) with \(\mathrm {fpr}(E\varGamma ,g)>1/3\). Assume that \(G_v^{\varGamma (v)}\) is cyclic of order 2.
For a vertex \(w \in V\varGamma \), let \(w'\) be the neighbour of w such that \(\{w'\}\) is the orbit of \(G_w\) of length 1. Then clearly \((w')' = w\) and \(G_w = G_{w'}\). Hence, the set \({\mathcal {M}} := \{\{w, w' \} \mid w \in V\varGamma \}\) is a complete matching of \(\varGamma \), while edges outside \({\mathcal {M}}\) form a 2factor \({\mathcal {F}}\). The group G in its action on \(E\varGamma \) fixes setwise both \({\mathcal {F}}\) and \({\mathcal {M}}\) and acts transitively on the arcs of each of these two sets. Let \({\tilde{\varGamma }}\) be the graph with vertexset \({\mathcal {M}}\) and two vertices \(e_1 , e_2 \in {\mathcal {M}}\) adjacent if and only if they are (as edges of \(\varGamma \)) at distance 1 in \(\varGamma \). The graph \({\tilde{\varGamma }}\) is then called the merge of \(\varGamma \). We may also think of \(\varGamma \) as being obtained by contracting all the edges in \({\mathcal {M}}\). The group G acts as an arctransitive group of automorphisms on \({\tilde{\varGamma }}\). Moreover, the connected components of the 2factor \({\mathcal {F}}\) gives rise to a decomposition \({\mathcal {C}}\) of \(E{\tilde{\varGamma }}\) into cycles.
Since we are assuming that \(\varGamma \) is neither a prism nor a Möbius ladder, [12, Lemma 9 and Theorem 10] implies that \({\tilde{\varGamma }}\) is 4valent. Moreover, the action of G on \({\tilde{\varGamma }}\) is faithful, arctransitive but not 2arctransitive.
Noticing that \(E{\tilde{\varGamma }}=2V{\tilde{\varGamma }}\), we can link the fixedpoint ratios of \(\varGamma \) and its 4valent merge \({\tilde{\varGamma }}\) as follows
Observe that either \(\mathrm {fpr}(V{\tilde{\varGamma }},g)> 1/3\) or \(\mathrm {fpr}(E{\tilde{\varGamma }},g)> 1/3\), otherwise
Using [11, Theorem 1.1] when \(\mathrm {fpr}(V{\tilde{\varGamma }},g)> 1/3\), and using Theorem 1 when \(\mathrm {fpr}(E{\tilde{\varGamma }},g)> 1/3\), it follows that either \({\tilde{\varGamma }}\cong C(r,s)\), with \(3s<2r\), or \(V{\tilde{\varGamma }}\le 70\). The latter case yields \(V\varGamma \le 140\) and the veracity of Theorem 2 follows with an inspection on the connected 3valent graphs having at most 140 vertices.
Therefore, we can suppose \({\tilde{\varGamma }}\cong C(r,s)\). In view of [12, Theorem 12], the graph \(\varGamma \) can be uniquely reconstructed from \({\tilde{\varGamma }}\) and the decomposition \({\mathcal {C}}\) of \(E{\tilde{\varGamma }}\) arising from the 2factor \({\mathcal {F}}\) via the splitting operation defined in Sect. 2.4. Hence, \(\varGamma \cong S(C(r,s))\). Finally, observe that
(The \(\tau _i\)’s are defined in Sect. 2.3.) Hence, a direct computation leads to \(3s<2r2\).
5.3 Proof of Theorem 2 when the local group is transitive
Let \(\varGamma \) be a finite connected 3valent vertextransitive graph, let \(v\in V\varGamma \), let \(G:=\mathrm {Aut}(\varGamma )\) and let \(g\in G\setminus \{1\}\) with \(\mathrm {fpr}(E\varGamma ,g)>1/3\). Assume that \(G_v^{\varGamma (v)}\) is transitive. Let \(s\ge 1\) such that G is sarctransitive and G is not \((s+1)\)arctransitive. Tutte’s theorem [19] implies that G is sarcregular.
Similarly to Definition 1, we partition \(E\varGamma \) with respect to the action of g.

We let \(A(\varGamma ,g)\) be the set of edges which are pointwise fixed by g, that is, \(\{a,b\}\in A(\varGamma ,g)\) if and only if \(\{a,b\}\in E\varGamma \), \(a^g=a\) and \(b^g=b\);

we let \(F(\varGamma ,g):=\mathrm {Fix}(E\varGamma ,g)\setminus A(\varGamma ,g)\), that is, \(\{a,b\}\in F(\varGamma ,g)\) if and only if \(\{a,b\}\in E\varGamma \), \(a^g=b\) and \(b^g=a\);

we let \(N(\varGamma ,g):=E\varGamma \setminus \mathrm {Fix}(E\varGamma ,g)\).
We let \(\varGamma [g]\) denote the subgraph of \(\varGamma \) induced by \(\varGamma \) on the vertices which are incident with edges in \(A(\varGamma ,g)\).^{Footnote 1} The edgeset of \(\varGamma [g]\) is \(A(\varGamma ,g)\) and its vertices are 1 or 3valent. Given \(i\in \{1,3\}\), we let \(V_i(\varGamma ,g)\) denote the set of vertices of \(\varGamma [g]\) having valency i.
Suppose \(\varGamma [g]\) is not a forest. Then \(\varGamma [g]\) contains an \(\ell \)cycle C. From Lemma 4, we have \(g(\varGamma [g]) \ge g(\varGamma ) \ge s + 1\) and hence, from C, we can extract an sarc \((v_0,v_1,\ldots ,v_{s1})\). As g fixes this sarc and as G is sarcregular, we deduce \(g=1\), which is a contradiction. Therefore \(\varGamma [g]\) is a forest. Before proceeding with the proof of Theorem 2, we prove a preliminary lemma.
Lemma 15
We have \(2F(\varGamma ,g)+3V_1(\varGamma ,g)+V_3(\varGamma ,g)\le V\varGamma \).
Proof
When \(s=1\), the arcregularity of G implies \(V_1(\varGamma ,g)=V_3(\varGamma ,g)=\emptyset \) and the proof immediately follows. Hence for the rest of the proof we may suppose \(s\ge 2\).
We let
Since \(V_1(\varGamma ,g),V_3(\varGamma ,g),{\mathcal {F}},{\mathcal {N}}\) are pairwise disjoint and since \({\mathcal {F}}=2F(\varGamma ,g)\), it suffices to show that \({\mathcal {N}}\ge 2V_1(\varGamma ,g)\). We divide this proof according to the girth of \(\varGamma \).
Suppose \(g(\varGamma )\ge 5\). Here, we construct an auxiliary graph \(\varDelta \). The vertex set of \(\varDelta \) is \(V_1(\varGamma ,g)\cup {\mathcal {N}}\) and we declare a vertex \(v\in V_1(\varGamma ,g)\) adjacent to a vertex \(u\in {\mathcal {N}}\) if \(\{v,u\}\in E\varGamma \). By construction, \(\varDelta \) is bipartite with parts \(V_1(\varGamma ,g)\) and \({\mathcal {N}}\). Given \(v\in V_1(\varGamma ,g)\), the automorphism g acts as a 2cycle on \(\varGamma (v)\). Let \(v_1,v_2\in \varGamma (v)\) forming the 2cycle of g. Then \(\{v,v_1\},\{v,v_2\}\in N(\varGamma ,g)\) and hence \(v_1,v_2\in {\mathcal {N}}\). This shows that each vertex in \(V_1(\varGamma ,g)\) has two neighbours in \({\mathcal {N}}\). As \(g(\varGamma )\ge 5\), we have \(g(\varDelta )\ge 5\) and hence \(2V_1(\varGamma ,g)\le {\mathcal {N}}\), because \(\varDelta (v)\cap \varDelta (v')=\emptyset \) for any two distinct vertices \(v,v'\in V_1(\varGamma ,g)\).
Suppose \(g(\varGamma )= 3\). Let \(\varGamma (v) = \{w_1 , w_2 , w_3\}\). Without loss of generality, suppose \(w_1\) and \(w_2\) are adjacent. Since G is arctransitive, \(w_i\) is adjacent to \(w_j\) for any \(i \ne j\). Thus \(\varGamma \cong K_4\). The graph \(K_4\) admits no nonidentity automorphisms with \(\mathrm {fpr}(E\varGamma ,g)>1/3\).
Suppose \(g(\varGamma )= 4\). Since \(s\ge 2\), [8, Theorem 1.1 and Table I] implies that \(\varGamma \) is isomorphic to either \(K_{3,3}\) or \(K_{4,4}4K_2\). In both cases, \(\varGamma \) does not admit a nonidentity automorphism g with \(\mathrm {fpr}(E\varGamma ,g)>1/3\). \(\square \)
We now resume our proof of Theorem 2. As \(2E\varGamma =3V\varGamma \), from Lemma 15, we have
Let c be the number of connected components of \(\varGamma [g]\). From Euler’s formula, we have \(V\varGamma [g]E\varGamma [g]=c\). Let \({\mathcal {S}}:=\{(v,w)\in V\varGamma [g]\times V\varGamma [g]\mid \{v,w\}\in E\varGamma [g]\}\). Then
It follows that \(V_3(\varGamma ,g)=V_1(\varGamma ,g)2c<V_1(\varGamma ,g)\). Using (5.1) and this inequality, we obtain \(\mathrm {fpr}(E\varGamma ,g)\le 1/3\), which is our final contradiction.
Data availability
The datasets analysed during the current study are available at https://www.fmf.unilj.si/~potocnik/work.htm.
Notes
During the revision process of this manuscript, we found out that \(\varGamma [g]\) has already been investigated in [6].
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Barbieri, M., Grazian, V. & Spiga, P. On the number of fixed edges of automorphisms of vertextransitive graphs of small valency. J Algebr Comb 57, 329–348 (2023). https://doi.org/10.1007/s10801022011765
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DOI: https://doi.org/10.1007/s10801022011765
Keywords
 Valency 3
 Valency 4
 Vertextransitive
 Arctransitive
 fixedpoints
Mathematics Subject Classification
 05C25
 20B25