On the number of fixed edges of automorphisms of vertex-transitive graphs of small valency

We prove that, if Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma $$\end{document} is a finite connected 3-valent vertex-transitive, or 4-valent vertex- and edge-transitive graph, then either Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma $$\end{document} is part of a well-understood family of graphs, or every non-identity automorphism of Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma $$\end{document} fixes at most 1/3 of the edges. This answers a question proposed by Primož Potočnik and the third author.


Introduction
Potočnik and Spiga have proved in [PS21b] that, if Γ is a finite connected 3-valent vertextransitive graph, or a 4-valent vertex-and edge-transitive graph then, unless Γ belongs to a well-known family of graphs, every non-identity automorphism of Γ 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 [PS21b,Problem 1.7].In this paper we solve this problem.
Theorem 1.1.Let Γ be a finite connected 4-valent vertex-and edge-transitive graph admitting a non-identity automorphism fixing more than 1/3 of the edges.Then one of the following holds: (1) Γ is isomorphic to the complete graph on 5 vertices; (2) Γ is isomorphic to a Praeger-Xu graph C(r, s), for some r and s with 3s < 2r − 3.
Theorem 1.2.Let Γ be a finite connected 3-valent vertex-transitive graph admitting a nonidentity automorphism fixing more than 1/3 of the edges.Then Γ is isomorphic to a Split Praeger-Xu graph SC(r, s), for some r and s with 3s < 2r − 3.
We refer to Section 2.3 for the definition of the ubiquitous Praeger-Xu graphs and for their splittings.The bound in Theorem 1.2 is sharp.For instance, each 3-valent graph admitting a non-identity automorphism fixing setwise a complete matching has the aforementioned property.For valency 4, we conjecture that the bound 1/3 in Theorem 1.1 can be strengthen to 1/4, by eventually including some more small exceptional graphs in part (1).
Theorem 1.1 and 1.2 rely on the following group-theoretic fact: Theorem 1.3 ([PS21a], Theorem 1.1).Let G be a finite transitive permutation group on Ω containing no non-identity normal subgroup of order a power of 2. Suppose there exists ω ∈ Ω such that the stabilizer G ω of ω in G is a 2-group.Then, every non-identity element of G fixes at most 1/3 of the points.
The main results of this paper and the results in [PS21b] show that, besides small exceptions or well-understood families of graphs, non-identity automorphisms of 3-valent or 4-valent 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 4-valent graphs, because of the assumption of edge-transitivity).In our opinion, the difficulty in having a unifying theory of vertex-transitive graphs of small valency admitting non-identity automorphisms fixing too many vertices or edges is due to our lack of understanding possible generalizations of Praeger-Xu graphs.That is, vertex-transitive 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 arc-transitive graphs, is in [LPS21].
Investigations on the number of fixed points of graph automorphisms do have interesting applications.For instance, very recently Potočnik, Toledo and Verret [PTV] pivoting on the results in [PS21b] have proved remarkable results on the cycle structure of general automorphisms of 3-valent vertex-transitive and 4-valent arc-transitive graphs.
1.1.Structure of the paper.In Section 2, we introduce some basic terminology and, in particular, we introduce the Praeger-Xu graphs and their splittings.Then, we start in Section 3 with some preliminary results.In Section 4, we prove Theorem 1.1 and, in Section 5, we prove Theorem 1.2.

The players
2.1.Basic group-theoretic notions.Given a permutation g on a set Ω, we write Fix(Ω, g) for the set of fixed points of g, i.e.
and we write fpr(Ω, g) for the fixed-point-ratio of g, i.e.
A permutation group G on Ω is said to be semiregular if the identity is the only element fixing some point.When G is semiregular and transitive on Ω, the group G is regular on Ω.
Given a permutation group G of Ω and a partition Σ of Ω, we say that Σ is G-invariant if σ g ∈ π, for every σ ∈ Σ.Given a normal subgroup N of G, the orbits of N on Ω form a G-invariant partition, which we denote by Ω/N .
We present here a useful lemma involving the notion just defined.

2.2.
Basic graph-theoretic notions.In this paper, a digraph is binary relation where AΓ ⊆ V Γ × V Γ.We refer to the elements of V Γ as vertices and to the elements of AΓ as arcs.A graph is a finite simple undirected graph, i.e. a pair where V Γ is a finite set of vertices, and EΓ is a set of unordered pairs of V Γ, 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 non-negative integer s, an s-arc of Γ is an ordered set of s + 1 adjacent vertices with any three consecutive elements pairwise distinct.When s = 0, an s-arc is simply a vertex of Γ; when s = 1, an s-arc is simply an arc, that is, an oriented edge.The girth of Γ, denoted by g(Γ), is the minimum length of a cycle in Γ.We denote by Γ(v) the neighbourhood of the vertex v.The size of |Γ(v)| is the valency of v.We are mainly dealing with regular graphs, that is, with graphs where |Γ(v)| is constant as v runs through the elements of V Γ.In these cases, we refer to the valency of the graph.
Let Γ be a graph, let G be a subgroup of the automorphism group Aut(Γ) of Γ, let v ∈ V Γ and let w ∈ Γ(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 Γ(v) and we denote by G the kernel of the action of G v on Γ(v).Now, the permutation group induced by and we have When G acts transitively on the set of s-arcs of Γ, we say that G is s-arc-transitive.When s = 0, we say that G is vertex-transitive and, when s = 1, we say that G is arc-transitive.Moreover, when G acts regularly on the set of s-arcs of Γ we emphasis this fact by saying that G is s-arc-regular.
When G acts transitively on EΓ, we say that G is edge-transitive.Finally, when G is edgeand vertex-transitive, but not arc-transitive, we say that G is half-arc-transitive.This name comes from the fact that G has two orbits on ordered pairs of adjacent vertices of Γ (a.k.a.arcs), each orbit containing precisely one of the two arcs underlying each edge.
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 Γ := Cay(G, S) having vertex set G and where (g, h) ∈ AΓ if and only if gh −1 ∈ G. Now, 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 | s ∈ S}.Observe that the right regular representation of G acts as a group of automorphisms on 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 4-valent Praeger-Xu graphs C(r, s) through their directed counterpart defined in [Pra89].Further details on Praeger-Xu graphs can be found in [GP94,JPW19,PX89].We also advertise [JPW22], 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 ≥ 3 be an integer.Then 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 C(r, 1) = Z r × Z 2 , and the two arcs starting in (x, i) end in (x + 1, 0) and in (x + 1, 1).For any 2 ≤ s ≤ r − 1, V C(r, s) is defined as the set of all (s − 1)-arcs of C(r, 1), and (v 0 , v 1 , . . ., v s−1 ) ∈ V C(r, s) is the beginning point of the two arcs ending in (v 1 , v 2 , . . ., v s−1 , u) and in (v 1 , v 2 , . . ., v s−1 , u ′ ), where u and u ′ are the two vertices of C(r, 1) that prolong the (s − 1)-arc (v 1 , v 2 , . . ., v s−1 ).The Praeger-Xu graph C(r, s) is then defined as the non-oriented underlying graph of C(r, s).It can be verified that C(r, s) is a connected 4-valent 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 Aut(C(r, 1)) on the set of s-arcs of C(r, 1).Let i ∈ Z r and let τ i be the transposition on V C(r, 1) swapping the vertices (i, 0) and (i, 1) and fixing the remaining vertices.Since τ i is an automorphism of C(r, 1), it is immediate to extended the action of τ 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 Praeger-Xu graphs, it is also natural to define on V C(r, 1) the permutations ρ and σ as follows (x, i) ρ = (x + 1, i), and (x, i) σ = (−x, i).
While ρ is an automorphism of C(r, 1), σ is an automorphism of C(r, 1) but not of V C(r, 1).Moreover, observe that the group ρ, σ normalizes K. Define H = K ρ, σ , and H + = K ρ , and, as for K, the symbols H and H + will always refer to these groups.Clearly H ∼ = C 2 ≀ D r is a group of automorphisms of C(r, s) and H + ∼ = C 2 ≀ C r is a group of automorphisms of C(r, s).
Moreover, H acts vertex-and edge-transitively on C(r, s) (and so does H + on C(r, s)), but not 2-arc-transitively.The Praeger-Xu graphs also admit the following algebraic characterization.
Lemma 2.3.Let Γ be a finite connected 4-valent graph and let G be a vertex-and edge-transitive group of automorphisms of Γ.If G has an abelian normal subgroup which is not semiregular on V Γ, then Γ is isomorphic to a Praeger-Xu graph C(r, s), for some integers r and s.
Proof.It follows by [Pra89, Theorem 2.9] and [PX89, Theorem 1] upon setting p = 2. 2.4.Split Praeger-Xu graphs.For our purposes, the Split Praeger-Xu graphs are obtained form the Praeger-Xu graphs via the splitting operation which was introduced in [PSV13, Construction 9], and which we will comment upon in Section 5.
Here we give an explicit description of SC(r, s).Split any vertex of C(r, s) into two copies, say v + and v − .For any arc of 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 C(r, s).

Preliminary results
3.1.Graph-theoretical considerations.In this section, we develop our tool box that extends outside the scope of proving our main theorems.
Lemma 3.1.Let Γ be a connected k-valent graph, with k ≥ 3, and let G be an s-arc-transitive group of automorphisms of Γ.Then the girth of Γ is greater than s, i.e. g(Γ) ≥ s + 1.
Proof.Let v 0 , v 1 , . . ., v ℓ−2 , v ℓ−1 be the vertices of a cycle of Γ of length ℓ, where v i is adjacent to v i+1 , for every i. (Computations are performed mod ℓ.)We argue by contradiction and we suppose ℓ ≤ s.Consider two s-arcs in Γ of the form As G is transitive on s-arcs and the previous s-arcs are not G-conjugate, we find a contradiction.

Lemma 3.2. Let Γ be a finite connected graph and let
Proof.Let u ∈ V Γ.As Γ is connected, we prove the existence of t u ∈ T with v tu = u arguing by induction on the minimal distance d := d(v, u) from v to u in Γ.When d = 0, that is, v = u, we may take t u to be the identity of T .Suppose then . By hypothesis, t u ′ ∈ T and v t ′ = u ′ .Therefore, v t u ′ t = u ′t = u and we may take t u := t u ′ t.A general result on the fixed-point-ratio of Cayley graphs can be proven regardless of the valency.
Lemma 3.4.Let G be a finite group, let S be an inverse closed non-empty subset of G, let Γ := Cay(G, S) and let g ∈ G \ {1}.If fpr(EΓ, g) = 0, then g 2 = 1 and In particular, fpr(EΓ, g) ≤ 1/|S| and the equality is attained if and only if g G ⊆ S.
Proof.Suppose fpr(EΓ, g) = 0. We let C G (g) denote the centralizer of g in G.
For each s ∈ S, let E s := {{x, sx} | x ∈ G}.Observe that E s is a complete matching of Γ and that {E s | s ∈ S} is a partition of the edge set EΓ.
Let s ∈ S. Suppose E s ∩ Fix(EΓ, g) = ∅ and fix {x, sx} ∈ E s ∩ Fix(EΓ, g).As g fixes the edge {x, sx}, we have xg = sx and sxg = x.We deduce g 2 = 1 and s = xg x−1 .In other words, g has order 2 and g has a conjugate in S. Now, for every {x, sx} ∈ E s , with a similar computation, we obtain that {x, sx} ∈ Fix(EΓ, g) if and only if The previous paragraph has established that g has order 2.Moreover, for each s ∈ S, E s ∩ Fix(EΓ, g) = ∅ if and only if s ∈ g G .Furthermore, in the case that s ∈ g G , the cardinality of E s ∩ Fix(EΓ, g) does not depend on s and equals |C G (g)|/2.Therefore, The next lemma studies the nature of fixed edges in a Praeger-Xu graph.
Suppose r = 4.By Lemma 2.2, Aut(Γ) = H = K ρ, σ .In particular, Denote by ∆ x the set of (s − 1)-arcs in C(r, 1) starting at (x, 0) or at (x, 1).From the definition of the vertex set of C(r, s), we have Moreover, ∆ x is a K-orbit and the subgraph induced by Γ on ∆ x ∪ ∆ x+1 is the disjoint union of cycles of length 4. Observe that, for any x ∈ Z r , We start by proving that g ∈ K.
Suppose ε = 1.Since ρ, σ 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 ∈ {0, 1}.
Assume i = 0. Let {a, b} ∈ Fix(EΓ, g).As above, replacing a with b if necessary, we may suppose that a ∈ ∆ x and b ∈ ∆ x+1 , for some x ∈ Z r .If a g = a and b g = b, we have ∆ g x = ∆ x and ∆ g x+1 = ∆ x+1 .Now, (3.1) yields −x − s + 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 ∆ g x = ∆ x+1 and ∆ g x+1 = ∆ x .Now, (3.1) yields −x − s + 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 Z r and, when r is even, the equation 2x + s = 0 has either zero or two solutions in Z r depending on whether s is odd or even.Recalling that the subgraph induced by Γ on ∆ x ∪ ∆ x+1 is a disjoint union of cycles of length 4, we obtain that In both cases, we have fpr(EΓ, g) ≤ 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} ∈ Fix(EΓ, g).As above, replacing a with b if necessary, we may suppose that a ∈ ∆ x and b ∈ ∆ x+1 , for some x ∈ Z r .If a g = a and b g = b, we have ∆ g x = ∆ x and ∆ g x+1 = ∆ 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 ∆ g x = ∆ x+1 and ∆ g x+1 = ∆ 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 Z r depending on whether s is even or odd.Recalling that the subgraph induced by Γ on ∆ x ∪ ∆ x+1 is a disjoint union of cycles of length 4, we obtain that Thus, we have fpr(EΓ, g) ≤ 1/4, which is a contradiction.
Since g ∈ K, if g fixes the edge {a, b} ∈ EΓ, then g fixes both end-vertices a and b.It remains to show that 3s < 2r − 3. Notice that τ i moves precisely those (s − 1)-arcs of C(r, 1) that pass through one of the vertices (i, 0) or (i, 1).Therefore, τ 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 τ i , such an element fixes at most as many edges as a single τ i .Hence Lemma 3.6.Let Γ = C(r, s) be a Praeger-Xu graph, let G be a vertex-and edge-transitive group of automorphism of Γ containing a non-identity element g fixing more that 1/3 of the edges and with G not 2-arc-transitive.Then G is Aut(Γ)-conjugate to a subgroup of H as defined in Section 2.3.
Lemma 3.7 ([PS21b], Lemma 1.11).Let Γ be a finite connected 4-valent graph, let G be a vertexand edge-transitive group of automorphisms of Γ, and let N be a minimal normal subgroup of G.
If N is a 2-group and Γ/N is a cycle of length at least 3, then Γ is isomorphic to a Praeger-Xu graph C(r, s) for some integers r and s.

Proof of Theorem 1.1
In this section we prove Theorem 1.1.Our proof is divided into two cases, depending on whether Γ admits a group of automorphisms acting 2-arc-transitively or not.4.1.Proof of Theorem 1.1 when Γ is 2-arc-transitive.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 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 anti-flags in the plane, i.e. the point p is adjacent to the line L if, and only if, p / ∈ L. The automorphism group of BCH is isomorphic to SL 3 (2).2.A non-identity automorphism of BHC fixes at most 4 edges out of 28.
We let Γ[g] denote the subgraph of Γ induced by Γ on the vertices which are incident with edges in A(Γ, g).The edge-set of Γ[g] is A(Γ, g) and its vertices are 1-, 2-or 4-valent.Given i ∈ {1, 2, 4}, we let V i (Γ, g) denote the set of vertices of Γ[g] having valency i.
Lemma 4.3.Let Γ be a finite connected 4-valent graph of girth g(Γ) ≥ 5 and let g be an automorphism of Γ.
Proof.We let We construct an auxiliary graph ∆.The vertex Given v ∈ V 1 (Γ, g), the automorphism g acts as a 3-cycle on Γ(v).Let v 1 , v 2 , v 3 ∈ Γ(v) forming the 3-cycle of g.Then {v, v 1 }, {v, v 2 }, {v, v 3 } ∈ N (Γ, g) and hence v 1 , v 2 , v 3 ∈ N .This shows that each vertex in V 1 (Γ, g) has three neighbours in N .Similarly, each vertex in V 2 (Γ, g) has two neighbours in N .As g(Γ) > 4, we have g(∆) > 4 and hence 3|V 1 (Γ, g An amalgam is a triplet (L, B, R) of groups such that B = L ∩ R, and its index is the couple (|L : B|, |R : B|).The amalgam (L, B, R) is said to be faithful if no subgroup of B is normal in L, R ; moreover, (L, B, R) is said to be 2-transitive if the action of L on the right cosets of B by right multiplication is 2-transitive.
Observe that, if Γ is a finite connected G-arc-transitive graph of valency k, then for any v ∈ V Γ and w ∈ Γ(v), the triplet Finite faithful 2-transitive amalgams of index (4, 2) have been studied in detail by Potočnik in [Pot08].We use this work to deduce some properties on Fix(EΓ, g).
Proof.If G is s-arc-regular, then g = 1 because g fixes an s-arc.Using [Pot08], we see that there are 6 amalgams such that G is not s-arc-regular.For each of these remaining amalgams a case-by-case computation shows that the only automorphism leaving the neighbourhood of each end of a given s-arc fixed is the identical map.Lemma 4.5.Let Γ be a finite connected 4-valent graph of girth g(Γ) ≥ 5, let G be a 2-arctransitive group of automorphisms of Γ such that G w is a 3-group, for any two distinct vertices at distance at most 2, and let g ∈ G \ {1}.
Proof.Assume that the vertices in V 4 (Γ, g) are at pairwise distance more than 2. Then any two such vertices share no common neighbour.In particular, v∈V4(Γ,g) Γ(v) has cardinality 4|V 4 (Γ, g)| and is contained in and the lemma immediately follows in this case.
Assume that there exist two distinct vertices v and w of V 4 (Γ, g) having distance at most 2. In particular, g ∈ G w and hence g has order a power of 3, because w is a 3-group.Observe that V 2 (Γ, g) = ∅ because an element of order 3 in a local group cannot fix exactly two elements.Let s ≥ 2 such that G is s-arc-transitive, but not (s + 1)-arc-transitive.
If g(Γ) ≤ 4, then the proof follows from Lemma 4.1 and from the remarks at the beginning of Section 4.1.Therefore, for the rest of the proof we suppose that g(Γ) > 4. Since 4|V Γ| = 2|EΓ|, we have where in the last inequality we have used Lemma 4.3.We claim that, for any two distinct vertices v, w ∈ V Γ at distance at most 2 one of the following holds The claim follows with a case-by-case computation on the finite faithful 2-transitive amalgams of index (4, 2) classified in [Pot08].We now divide the proof according to (i) and (ii).
Let Γ be a finite connected vertex-and edge-transitive 4-valent graph admitting a non-identity automorphism g fixing more than 1/3 of the edges and with G := Aut(Γ) not 2-arc-transitive.If Γ is isomorphic to a Praeger-Xu graph, then part (2) of Theorem 1.1 holds.Therefore, for the rest of the argument, we suppose that Γ is not isomorphic to C(r, s), for any choice of r and s with r ≥ 3 and 1 is not 2-transitive on Γ(v).Since G is vertex-and edge-transitive, we obtain that either G has two orbits of cardinality 2. In both cases, we deduce that G If G has no non-identity normal subgroups having cardinality a power of 2, Theorem 1.3 (applied to the faithful and transitive action of G on EΓ) contradicts fpr(EΓ, g) > 1/3.Thus, G has a minimal normal 2-subgroup N .
As Γ is not isomorphic to a Praeger-Xu graph, Lemma 2.3 yields that N acts semi-regularly on V Γ.Consider the quotient graph Γ/N and observe that, as G is vertex-and edge-transitive, Γ/N has valency 0, 1, 2 or 4.
If Γ/N has valency 0, then N is transitive on V Γ.Thus N is vertex-regular on Γ.As Γ is connected of valency 4, N is generated by at most 4 elements and hence |V Γ| = |N | divides 2 4 .If Γ/N has valency 1, then N has two orbits on V Γ.Moreover, [PS21b, Lemma 1.14] implies that |V Γ| = 2|N | divides 128.In both cases, the statement can be checked computationally by inspecting the candidate graphs from the census of all 4-valent vertex-and edge-transitive graphs of small order, see [PSV13,PSV15].If Γ/N has valency 2, then we contradict Lemma 3.7.Therefore, for the rest of the proof, we may suppose that Γ/N has valency 4.
Observe that G/N acts faithfully as a group of automorphisms on Γ/N .Moreover, G/N acts vertex-and edge-transitively on Γ/N , but not 2-arc-transitively.Observe that g / ∈ N , because the elements in N fix no edge of Γ.Thus gN is not the identity automorphism of Γ/N and, by Lemma 2.1, we have fpr(EΓ/N, N g) > 1/3.Our inductive hypothesis on |V Γ| implies that Γ/N is isomorphic to K 5 or to a Praeger-Xu graph C(r, s) with 3s < 2r − 3.
Assume Γ/N ∼ = K 5 .Now, Aut(K 5 ) = S 5 and S 5 contains a unique conjugacy class of subgroups which are vertex-and edge-transitive, but not 2-transitive (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 | ≤ 16.We deduce |V Γ| ≤ 10 • 16 = 160 and, as above, the statement can be checked computationally by inspecting the census of all 4-valent vertex-and edge-transitive graphs of small order.
Assume Γ/N ∼ = C(r, s), for some r and s with 3s < 2r − 3. From Lemma 3.6, G/N is Aut(Γ)conjugate to a subgroup of H as defined in Section 2.3.Without loss of generality, we can identify G/N with such subgroup, so that G/N ≤ H. Now, we first deal with the exceptional case (r, s) = (4, 1).As G/N is a 2-group and N is a minimal normal subgroup of G, we deduce |N | = 2 and hence |V Γ| = |V Γ/N ||N | = 4 • 2 = 8.Now, the proof follows inspecting the vertexand edge-transitive graphs of order 8. Therefore, for the rest of the argument, we suppose (r, s) = (4, Now, Lemma 3.5 implies N g ∈ K ≤ H + .Denote by X the group G/N ∩ H + .This group is an half-arc-transitive group of automorphisms of Γ/N and, since |H : H + | = 2, we have |G/N : X| ≤ 2. Denote by G + the preimage of X with respect to the quotient projection G → G/N , so that G + /N ∼ = X.Now, G + acts half-arc-transitively on Γ and, from N g ∈ X, we see that g ∈ 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 ≤ H + .By Lemma 3.5, all the edges fixed in Γ/N by gN are fixed as arcs.Therefore, all the edges fixed in Γ by g are fixed as arcs.

Proof of Theorem 1.2
We now turn our attention to finite connected 3-valent vertex-transitive graphs.We divide the proof of Theorem 1.2 in three cases, which we now describe.Let Γ be a finite connected 3-valent vertex-transitive graph, let G := Aut(Γ) and let v ∈ V Γ.The local group G Γ(v) v is a subgroup of the symmetric group of degree 3 and we divide the proof of Theorem 1.2 depending on the structure of G = 1, the connectivity of Γ implies G v = 1 and hence G acts regularly on V Γ.In this case an observation of Sabidussi [Sab58] yields that Γ is Cayley graph over G.We deal with this case in Section 5.1.When G Γ(v) v is cyclic of order 2, [PSV13] has established a fundamental relation between Γ and a certain finite connected 4-valent graph; in Section 5.2, we exploit this relation and Theorem 1.1 to deal with this case.When G Γ(v) v is transitive, Γ is arc-transitive and we use the amazing result of Tutte concerning the structure of G v to deal with this case in Section 5.3.5.1.Proof of Theorem 1.2 when the local group is the identity.Let Γ be a finite connected 3-valent vertex-transitive graph, let v ∈ V Γ, let G := Aut(Γ) and let g ∈ G \ {1}.Assume that G Γ(v) v 1. Lemma 3.4 yields fpr(EΓ, g) ≤ 1/3 and hence Theorem 1.2 holds in this case.5.2.Proof of Theorem 1.2 when the local group is cyclic of order 2. In our proof of this case, we need to refer to two families of 3-valent Cayley graphs.Given n ∈ N with n ≥ 3, the prim Pr n is the Cayley graph Pr n = Cay (Z n × Z 2 , {(0, 1), (1, 0), (−1, 0)}) .
For these two classes of graphs the proof of Theorem 1.2 follows with a computation.When n = 4, the automorphism group of Pr n is isomorphic to D n × C 2 and, for each x ∈ Aut(Pr n ) with x = 1, it can be verified that fpr(EPr n , x) ≤ 1/3, see also Lemma 3.4.The case n = 4 is exceptional, because Pr 4 ∼ = Q 4 is 2-arc-transitive and hence Pr 4 is of no concern to us here.Similarly, when n / ∈ {2, 3}, the automorphism group of Mb n is isomorphic to D 2n and, for each x ∈ Aut(Mb n ) with x = 1, it can be verified that fpr(EMb, x) ≤ 1/3, again see also Lemma 3.4.The cases n ∈ {2, 3} are exceptional, because Mb 2 ∼ = K 4 and Mb 3 are 2-arc-transitive and hence are of no concern to us here.Now, let Γ be a finite connected 3-valent vertex-transitive graph not isomorphic to Pr n and not isomorphic to Mb n , let v ∈ V Γ, let G := Aut(Γ) and let g ∈ G \ {1} with fpr(EΓ, g) > 1/3.Assume that G Γ(v) v is cyclic of order 2. and hence v 1 , v 2 ∈ N .This shows that each vertex in V 1 (Γ, g) has two neighbours in N .As g(Γ) ≥ 5, we have g(∆) ≥ 5 and hence 2|V 1 (Γ, g)| ≤ |N |, because ∆(v) ∩ ∆(v ′ ) = ∅ for any two distinct vertices v, v ′ ∈ V 1 (Γ, g).