Normal quotients of diameter at most two of finite three-geodesic-transitive graphs

Abstract

An s-geodesic of a graph is a path of length s such that the first and last vertices are at distance s. We study finite graphs Γ of diameter at least 3 for which some subgroup G of automorphisms is transitive on the set of s-geodesics for each $s\le 3$. If Γ has girth at least 6 then all 3-arcs are 3-geodesics so Γ is 3-arc-transitive, and such graphs have already been studied fruitfully; also graphs of girth 3 with these properties have been investigated successfully. We therefore focus on those of girth 4 or 5. We study their normal quotients ${\mathrm{\Gamma }}_N$ modulo the orbits of a normal subgroup N of G and prove that, provided ${\mathrm{\Gamma }}_N$ has diameter at least 3, then Γ is a cover of ${\mathrm{\Gamma }}_N$ and ${\mathrm{\Gamma }}_N,G/N$ have the same girth and transitivity properties as $\mathrm{\Gamma },G$ (so if $N\ne 1$ we may reduce consideration to a smaller graph in the family).

We then focus on the ‘degenerate case’ where ${\mathrm{\Gamma }}_N$ has diameter at most 2. In these cases also, Γ is a cover of ${\mathrm{\Gamma }}_N$ provided N has at least three vertex-orbits. If ${\mathrm{\Gamma }}_N$ is a complete graph ${\mathrm{K}}_r$ (diameter 1), then we prove that Γ is either the complete bipartite graph ${\mathrm{K}}_{r,r}$ with the edges of a perfect matching removed, or a unique 6-fold-cover of ${\mathrm{K}}_7$. In the remaining case where ${\mathrm{\Gamma }}_N$ has diameter 2, then ${\mathrm{\Gamma }}_N$ is a 2-arc-transitive strongly regular graph. We classify all the 2-arc-transitive strongly regular graphs, and using this classification we describe all their finite $(G,3)$-geodesic-transitive covers of girth 4 or 5, except for a few difficult cases.

Introduction

A geodesic from a vertex u to a vertex v in a graph Γ is a path of the shortest length from u to v in Γ, and is called an s-geodesic if the distance between u and v is s. If G is some subgroup of automorphisms of Γ, then Γ is said to be $(G,s)$-geodesic-transitive if it has an s-geodesic, and for each $i\le s$, G is transitive on the set of i-geodesics of Γ. In this paper all graphs are assumed to be finite. The systematic investigation of s-geodesic-transitive graphs was initiated recently. The possible local structures of 2-geodesic-transitive graphs were determined in [14]. Then Devillers, Li and the authors [15] classified 2-geodesic-transitive graphs of valency 4. Later, in [16], a reduction theorem for the family of normal 2-geodesic-transitive Cayley graphs was proved and those which are complete multipartite graphs were also classified. Our focus in this paper is on 3-geodesic-transitive graphs.

For a positive integer s, an s-arc of Γ is a sequence of vertices $(v_0,v_1,\dots ,v_s)$ in Γ such that $v_i,v_{i+1}$ are adjacent and $v_{j-1}\ne v_{j+1}$ where $0\le i\le s-1$ and $1\le j\le s-1$. In particular, 1-arcs are called arcs. If G is some subgroup of automorphisms of Γ, then Γ is said to be $(G,s)$-arc-transitive if, for each $i\le s$, G is transitive on the set of i-arcs of Γ. Study of s-arc-transitive graphs originates from work of Tutte [35], [36], who proved that there are no finite 6-arc-transitive cubic graphs, and that for such graphs the order of the stabiliser of a vertex is at most 48. This seminal result launched the study of s-arc-transitive graphs. About twenty years later, relying on the classification of finite 2-transitive groups (which in turn depends on the finite simple group classification), Weiss [37] proved that there are no finite 8-arc-transitive graphs with valency at least three. Moreover, for each $s\le 5$ and for $s=7$, there are graphs which are s-arc-transitive but not $(s+1)$-arc-transitive. However the situation for $s=6$ is different since each 6-arc-transitive graph is 7-arc-transitive, see [37, Theorem]. Many other results have been proved about s-arc-transitive graphs, see for example [1], [25], [29].

On the other hand, there is no upper bound on s for s-geodesic-transitivity [26, Theorem 1.1]. Clearly, every s-geodesic is an s-arc, but some s-arcs may not be s-geodesics, even for small values of s. If Γ has girth 3 (the girth of Γ is the length of the shortest cycle in Γ), then 2-arcs contained in 3-cycles are not 2-geodesics. If Γ has girth 4 or 5, then 3-arcs contained in 4-cycles or 5-cycles are not 3-geodesics. The graph in Fig. 1 is the Hamming graph $\mathrm{H}(3,2)$, and is $(G,3)$-geodesic-transitive but not $(G,3)$-arc-transitive with valency 3 and girth 4 where G is the automorphism group. Thus the family of $(G,3)$-arc-transitive graphs is properly contained in the family of $(G,3)$-geodesic-transitive graphs.

We are interested in s-geodesic-transitive graphs that are not s-arc-transitive. Such graphs for $s=2$ have been studied extensively, see for example [14], [15], [16], [17]. For $s=3$, the valency 4 examples have been classified in [26], and it was shown there that examples exist with unboundedly large diameter and valency. We make some comments in Remark 1.1 on the general context of our investigation.

Remark 1.1

Let $s\ge 2$ and let Γ be an s-geodesic-transitive graph that is not s-arc-transitive. Then Γ is not a cycle so Γ has valency at least 3. Moreover, Γ must contain some s-arcs that are not s-geodesics, and so Γ contains cycles of length at most $2s-1$. Thus the girth g of Γ, that is, the length of the shortest cycles in Γ, satisfies $g\le 2s-1$. If $g\le 2s-3$, then $(s-1)$-arcs in a g-cycle are not $(s-1)$-geodesics and hence Γ is $(s-1)$-geodesic-transitive but not $(s-1)$-arc-transitive. Thus for the study of such graphs the most interesting values for the girth g are $2s-1$ and $2s-2$.

If this holds and if $s\ge 9$, then $g\ge 2s-2\ge 16$, and hence each 8-arc is an 8-geodesic. This implies that Γ is 8-arc-transitive, contradicting Weiss' Theorem [37]. Thus graphs which are s-geodesic-transitive but not s-arc-transitive and which have girth $2s-1$ or $2s-2$ can exist only for $s\le 8$. Moreover, if $s=7$ and $g=12$ or 13, then each 6-arc is a 6-geodesic and so Γ is 6-arc-transitive. This implies that Γ is 7-arc-transitive by [37, Theorem], a contradiction. Hence in addition $s\ne 7$. As discussed above these graphs for $s=2$ have been well-studied, and the case $s=3$ is the subject of this paper. It would be interesting to know if similar analyses could be carried out successfully for the remaining values of s, namely $s\in \{4,5,6,8\}$, and in particular, whether this could be done for those with normal quotients of diameter less than s.

Problem 1.2

For $s\in \{4,5,6,7,8\}$, classify the finite $(G,s)$-geodesic-transitive graphs of girth $2s-1$ or $2s-2$, which are not $(G,s)$-arc-transitive.

We focus on the case $s=3$ in this paper. We introduce a general framework for describing all the graphs with these properties, via studying their normal quotients. Let Γ be a G-vertex-transitive graph. If N is a vertex-intransitive normal subgroup of G, then the quotient graph ${\mathrm{\Gamma }}_N$ of Γ is the graph whose vertex set is the set of N-orbits, such that two N-orbits $B_i,B_j$ are adjacent in ${\mathrm{\Gamma }}_N$ if and only if there exist $x\in B_i,y\in B_j$ such that $x,y$ are adjacent in Γ. Such quotients ${\mathrm{\Gamma }}_N$ are often referred to as G-normal quotients of Γ relative to N. Sometimes Γ is a cover of ${\mathrm{\Gamma }}_N$, that is to say, for each edge $\{B_i,B_j\}$ of ${\mathrm{\Gamma }}_N$ and $v\in B_i$, v is adjacent to exactly one vertex in $B_j$. In this case we say that Γ is a G-normal cover of ${\mathrm{\Gamma }}_N$ relative to N.

A connected regular graph is said to be strongly regular with parameters $(n,k,a,c)$ if it has n vertices, valency k, and if every pair of adjacent vertices has a common neighbours, and every pair of non-adjacent vertices has c common neighbours.

Our first theorem is a reduction result on the family of 3-geodesic-transitive graphs of girth 4 or 5. It describes the various possibilities for the girth and diameter of normal quotients. Note that 3-geodesic-transitive graphs have diameter at least 3 by definition, and therefore such graphs of girth at most 5 are not 3-arc-transitive.

Theorem 1.3

Let Γ be a connected $(G,3)$-geodesic-transitive graph of girth 4 or 5. Let N be a normal subgroup of G with at least 3 orbits on the vertex set. Then Γ is a cover of ${\mathit{\Gamma }}_N$, ${\mathit{\Gamma }}_N$ is $(G/N,s^{\prime })$-geodesic-transitive where $s^{\prime }=\min \{3,\mathrm{diam}({\mathit{\Gamma }}_N)\}$, and one of the following holds:

• ${\mathit{\Gamma }}_N$ is a complete graph.

• ${\mathit{\Gamma }}_N$ is a $(G/N,2)$-arc-transitive strongly regular graph with girth 4 or 5.

• ${\mathit{\Gamma }}_N$ has diameter at least 3 and the same girth as Γ.

The normal quotient graphs in Theorem 1.3(2) are 2-arc-transitive strongly regular graphs. In order to classify 3-geodesic-transitive graphs of girth 4 or 5, we need to know the 2-arc-transitive strongly regular graphs, and our second theorem determines all such graphs.

Theorem 1.4

Let Γ be a 2-arc-transitive strongly regular graph. Then either

• Γ has girth 4 and is one of the following graphs: ${\mathrm{K}}_{m,m}$ with $m\ge 2$, the Higman-Sims graph, the Gewirtz graph, the $M_{22}$-graph, or the folded 5-cube ${\square }_5$; or

• Γ has girth 5 and is one of the following graphs: $C_5$, the Petersen graph, or the Hoffman-Singleton graph.

Our third theorem, using the classification of 2-arc-transitive strongly regular graphs in Theorem 1.4, characterises all the $(G,3)$-geodesic-transitive covers Γ when ${\mathrm{\Gamma }}_N$ is in part (1) or (2) of Theorem 1.3. For each possible quotient ${\mathrm{\Gamma }}_N$, we obtain all normal covers Γ explicitly, except for three cases for ${\mathrm{\Gamma }}_N$ where we can only classify the graphs when Γ is $(G,4)$-distance-transitive (see Table 2).

Theorem 1.5

Let Γ be a connected $(G,3)$-geodesic-transitive graph of girth 4 or 5. Suppose that G has a normal subgroup N such that N has at least 3 orbits on vertices and ${\mathit{\Gamma }}_N$ has diameter at most 2. Then Γ is a cover of ${\mathit{\Gamma }}_N$ and either $\mathit{\Gamma },{\mathit{\Gamma }}_N$ are as in Table 1 or Γ is a non-$(G,4)$-distance-transitive graph of diameter at least 4 and ${\mathit{\Gamma }}_N$ is as in Table 2.

The graphs Γ and ${\mathrm{\Gamma }}_N$ in Table 1, Table 2 will be described in Subsection 2.2. In particular, the standard double cover is defined in Definition 2.4, and RGD denotes a resolvable divisible design. We remark that in Theorem 1.5, if ${\mathrm{\Gamma }}_N$ is strongly regular, then Γ and ${\mathrm{\Gamma }}_N$ may have distinct girths. For example, the Armanios-Wells graph is $(G,3)$-geodesic-transitive of girth 5 and it is a cover of the folded 5-cube ${\square }_5$ which has girth 4. Note that the cycle $C_r$, $r=4$ or 5, is a 2-arc-transitive graph of diameter 2. If Γ is a 3-geodesic-transitive cover of $C_r$, then Γ has valency 2, and hence Γ is a cycle with girth at least 8, and so is 3-arc-transitive.

In Theorem 1.3, we could choose N to be an intransitive normal subgroup of G which is maximal with respect to having at least 3 orbits on the vertex set of Γ. Then each non-trivial normal subgroup M of G, properly containing N, has 1 or 2 orbits, and so $M/N$ has 1 or 2 orbits on the vertices of ${\mathrm{\Gamma }}_N$. In other words, $G/N$ is quasiprimitive or bi-quasiprimitive on the vertex set of ${\mathrm{\Gamma }}_N$. Hence this theorem suggests that to investigate the family of $(G,3)$-geodesic-transitive graphs that are not $(G,3)$-arc-transitive, we should concentrate on the following two problems:

Problem 1.6

• determine $(G,3)$-geodesic-transitive graphs of girth 4 or 5 where G acts quasiprimitively or bi-quasiprimitively on the vertex set;

• investigate $(G,3)$-geodesic-transitive covers of the graphs obtained from (1), and also investigate the graphs Γ in Table 2.

This paper is organised as follows. After this introduction, we give, in Section 2, some definitions on groups and graphs that we need and also prove some elementary lemmas which will be used in the following analysis. Theorem 1.3 is proved in Section 3. It reduces the study of 3-geodesic-transitive graphs of girth 4 or 5 to the study of G-normal covers of complete graphs, of strongly regular graphs, and of a class of 3-geodesic-transitive graphs of diameter at least 3 with the same girth. Then in Section 4, we determine all the 2-arc-transitive strongly regular graphs and complete the proof of Theorem 1.4. In Section 5, we prove Theorem 1.5, that is, we investigate 3-geodesic-transitive graphs of girth 4 or 5 which are covers of a 2-arc-transitive graph that is complete or strongly regular.