On the radius and the attachment number of tetravalent half-arc-transitive graphs

TL;DR

This paper investigates the relationship between radius and attachment number in tetravalent half-arc-transitive graphs, proving specific divisibility properties and characterizing certain graph classes.

Contribution

It establishes that if the attachment number is twice an odd number, then it divides twice the radius, and characterizes graphs with radius 3 and attachment number 2 as certain covers of line graphs.

Findings

If a is twice an odd number, then a divides 2r.

Graphs with r=3 and a=2 are non-sectional split 2-fold covers of line graphs of 2-arc-transitive cubic graphs.

Confirmed that all graphs with a not dividing r are arc-transitive in the studied cases.

Abstract

In this paper, we study the relationship between the radius $r$ and the attachment number $a$ of a tetravalent graph admitting a half-arc-transitive group of automorphisms. These two parameters were first introduced in~[{\em J.~Combin.~Theory Ser.~B} {73} (1998), 41--76], where among other things it was proved that $a$ always divides $2r$. Intrigued by the empirical data from the census~[{\em Ars Math.\ Contemp.} {8} (2015)] of all such graphs of order up to 1000 we pose the question of whether all examples for which $a$ does not divide $r$ are arc-transitive. We prove that the answer to this question is positive in the case when $a$ is twice an odd number. In addition, we completely characterize the tetravalent graphs admitting a half-arc-transitive group with $r = 3$ and $a=2$, and prove that they arise as non-sectional split $2$-fold covers of line graphs of $2$-arc-transitive cubic graphs.

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1. Introduction

This paper stems from our research of finite simple connected tetravalent graphs that admit a group of automorphisms acting transitively on vertices and edges but not on the arcs of the graph; such groups of automorphisms are said to be half-arc-transitive. Observe that the full automorphism group $\mathrm{Aut}(\Gamma)$ of such a graph $\Gamma$ is then either arc-transitive or itself half-arc-transitive. In the latter case the graph $\Gamma$ is called half-arc-transitive.

Tetravalent graphs admitting a half-arc-transitive group of automorphisms are surprisingly rich combinatorial objects with connections to several other areas of mathematics (see, for example, [1, 9, 10, 11, 13, 16, 18]). One of the most fruitful tools for analysing the structure of a tetravalent graph $\Gamma$ admitting a half-arc-transitive group $G$ is to study a certain $G$-invariant decomposition of the edge set $E(\Gamma)$ of $\Gamma$ into the $G$-alternating cycles of some even length $2r$; the parameter $r$ is then called the $G$-radius and denoted $\mathop{{\rm rad}}_{G}(\Gamma)$ (see Section 2 for more detailed definitions). Since $G$ is edge-transitive and the decomposition into $G$-alternating cycles is $G$-invariant, any two intersecting $G$-alternating cycles meet in the same number of vertices; this number is then called the attachment number and denoted $\mathop{{\rm att}}_{G}(\Gamma)$. When $G=\mathrm{Aut}(\Gamma)$ the subscript $G$ will be omitted in the above notation.

It is well known and easy to see that $\mathop{{\rm att}}_{G}(\Gamma)$ divides $2\mathop{{\rm rad}}_{G}(\Gamma)$. However, for all known tetravalent half-arc-transitive graphs the attachment number in fact divides the radius. This brings us to the following question that we would like to propose and address in this paper:

Question 1**.**

Is it true that the attachment number $\mathop{{\rm att}}(\Gamma)$ of an arbitrary tetravalent half-arc-transitive graph $\Gamma$ divides the radius $\mathop{{\rm rad}}(\Gamma)$?

By checking the complete list of all tetravalent half-arc-transitive graphs on up to $1000$ vertices (see [15]), we see the that answer to the above question is affirmative for the graphs in that range. Further, as was proved in [14, Theorem 1.2], the question has an affirmative answer in the case $\mathop{{\rm att}}(\Gamma)=2$. In Section 3, we generalise this result by proving the following theorem.

Theorem 2**.**

Let $\Gamma$ be a tetravalent half-arc-transitive graph. If its radius $\mathop{{\rm rad}}(\Gamma)$ is odd, then $\mathop{{\rm att}}(\Gamma)$ divides $\mathop{{\rm rad}}(\Gamma)$. Consequently, if $\mathop{{\rm att}}(\Gamma)$ is not divisible by $4$, then $\mathop{{\rm att}}(\Gamma)$ divides $\mathop{{\rm rad}}(\Gamma)$.

As a consequence of our second main result (Theorem 3) we see that, in contrast to Theorem 2, there exist infinitely many arc-transitive tetravalent graphs $\Gamma$ admitting a half-arc-transitive group $G$ with $\mathop{{\rm rad}}_{G}(\Gamma)=3$ and $\mathop{{\rm att}}_{G}(\Gamma)=2$. In fact, in Section 2, we characterise these graphs completely and prove the following theorem (see Section 2.2 for the definition of the dart graph).

Theorem 3**.**

Let $\Gamma$ be a connected tetravalent graph. Then $\Gamma$ is $G$-half-arc-transitive for some $G\leq\mathrm{Aut}(\Gamma)$ with $\mathop{{\rm rad}}_{G}(\Gamma)=3$ and $\mathop{{\rm att}}_{G}(\Gamma)=2$ if and only if $\Gamma$ is the dart graph of some $2$-arc-transitive cubic graph.

The third main result of this paper, stemming from our analysis of the situation described by Theorem 3, reveals a surprising connection to the theory of covering projections of graphs. This theory has become one of the central tools in the study of symmetries of graphs. A particularly thrilling development started with the seminal work of Malnič, Nedela and Škoviera [5] who analysed the condition under which a given automorphism group of the base graph lifts along the covering projection. Recently, the question of determining the structure of the lifted group received a lot of attention (see [2, 6, 7]).

To be more precise, let $\wp\colon\tilde{\Gamma}\to\Gamma$ be a covering projection of connected graphs and let $\mathrm{CT}(\wp)$ be the corresponding group of covering transformations (see [5], for example, for the definitions pertaining to the theory of graph covers). Furthermore, let $G\leq\mathrm{Aut}(\Gamma)$ be a subgroup that lifts along $\wp$. Then the lifted group $\tilde{G}$ is an extension of $\mathrm{CT}(\wp)$ by $G$. If this extension is split then the covering projection $\wp$ is called $G$-split. The most natural way in which this can occur is that there exists a complement $\bar{G}$ of $\mathrm{CT}(\wp)$ in $\tilde{G}$ and a $\bar{G}$-invariant subset $S$ of $V(\tilde{\Gamma})$, that intersects each fibre of $\wp$ in exactly one vertex. In such a case we say that $S$ is a section for $\bar{G}$ and that $\bar{G}$ is a sectional complement of $\mathrm{CT}(\wp)$. Split covering projections without any sectional complement are called non-sectional. These turn out to be rather elusive and hard to analyse. To the best of our knowledge, the only known infinite family of non-sectional split covers was presented in [2, Section 4]. This family of non-sectional split covers involves cubic arc-transitive graphs of extremely large order.

In this paper we show that each connected tetravalent graph $\Gamma$ admitting a half-arc-transitive group $G$ of automorphisms such that $\mathop{{\rm att}}_{G}(\Gamma)=2$ and $\mathop{{\rm rad}}_{G}(\Gamma)=3$ is a $2$-fold cover of the line graph of a cubic $2$-arc-transitive graph, and that in the case when $\Gamma$ is not bipartite the corresponding covering projection is non-sectional. This thus provides a new and rather simple infinite family of the somewhat mysterious case of non-sectional split covering projections (see Section 4 for more details).

2. Half-arc-transitive group actions on graphs

In the next two paragraphs we briefly review some concepts and results pertaining half-arc-transitive group actions on tetravalent graphs that we shall need in the remainder of this section. For more details see [8], where most of these notions were introduced.

A tetravalent graph $\Gamma$ admitting a half-arc-transitive (that is vertex- and edge- but not arc-transitive) group of automorphisms $G$ is said to be $G$-half-arc-transitive. The action of $G$ induces two paired orientations of the edges of $\Gamma$ and for any one of them each vertex of $\Gamma$ is the head of two and the tail of the other two of its incident edges. (The fact that the edge $uv$ is oriented from $u$ to $v$ will be denoted by $u\to v$.) A cycle of $\Gamma$ for which every two consecutive edges either have a common head or common tail with respect to this orientation is called a $G$-alternating cycle. Since the action of $G$ is vertex- and edge-transitive all of the $G$-alternating cycles have the same even length $2\mathop{{\rm rad}}_{G}(\Gamma)$ and any two non-disjoint $G$-alternating cycles intersect in the same number $\mathop{{\rm att}}_{G}(\Gamma)$ of vertices. These intersections, called the $G$-attachment sets, form an imprimitivity block system for the group $G$. The numbers $\mathop{{\rm rad}}_{G}(\Gamma)$ and $\mathop{{\rm att}}_{G}(\Gamma)$ are called the $G$-radius and $G$-attachment number of $\Gamma$, respectively. If $G=\mathrm{Aut}(\Gamma)$ we suppress the prefix and subscript $\mathrm{Aut}(\Gamma)$ in all of the above definitions.

It was shown in [8, Proposition 2.4] that a tetravalent $G$-half-arc-transitive graph $\Gamma$ has at least three $G$-alternating cycles unless $\mathop{{\rm att}}_{G}(\Gamma)=2\mathop{{\rm rad}}_{G}(\Gamma)$ in which case $\Gamma$ is isomorphic to a particular Cayley graph of a cyclic group (and is thus arc-transitive). Moreover, in the case that $\Gamma$ has at least three $G$-alternating cycles, $\mathop{{\rm att}}_{G}(\Gamma)\leq\mathop{{\rm rad}}_{G}(\Gamma)$ holds and $\mathop{{\rm att}}_{G}(\Gamma)$ divides $2\mathop{{\rm rad}}_{G}(\Gamma)$. In addition, the restriction of the action of $G$ to any $G$-alternating cycle is isomorphic to the dihedral group of order $2\mathop{{\rm rad}}_{G}(\Gamma)$ (or to the Klein 4-group in the case of $\mathop{{\rm rad}}_{G}(\Gamma)=2$) with the cyclic subgroup of order $\mathop{{\rm rad}}_{G}(\Gamma)$ being the subgroup generated by a two-step rotation of the $G$-alternating cycle in question. In addition, if $C=(v_{0},v_{1},\ldots,v_{2r-1})$ is a $G$-alternating cycle of $\Gamma$ with $r=\mathop{{\rm rad}}_{G}(\Gamma)$ and $C^{\prime}$ is the other $G$-alternating cycle of $\Gamma$ containing $v_{0}$ then $C\cap C^{\prime}=\{v_{i\ell}\colon 0\leq i<a\}$ where $a=\mathop{{\rm att}}_{G}(\Gamma)$ and $\ell=2r/a$ (see [8, Proposition 2.6] and [12, Proposition 3.4]).

As mentioned in the Introduction one of the goals of this paper is to characterize the tetravalent $G$-half-arc-transitive graphs $\Gamma$ with $\mathop{{\rm rad}}_{G}(\Gamma)=3$ and $\mathop{{\rm att}}_{G}(\Gamma)=2$. The bijective correspondence between such graphs and $2$-arc-transitive cubic graphs (see Theorem 3) is given via two pairwise inverse constructions: the graph of alternating cycles construction and the dart graph construction. We first define the former.

2.1. The graph of alternating cycles

Let $\Gamma$ be a tetravalent $G$-half-arc-transitive graph for some $G\leq\mathrm{Aut}(\Gamma)$. The graph of $G$-alternating cycles $\mathrm{Alt}_{G}(\Gamma)$ is the graph whose vertex set consists of all $G$-alternating cycles of $\Gamma$ with two of them being adjacent whenever they have at least one vertex in common. We record some basic properties of the graph $\mathrm{Alt}_{G}(\Gamma)$.

Proposition 4**.**

Let $\Gamma$ be a connected tetravalent $G$-half-arc-transitive graph for some $G\leq\mathrm{Aut}(\Gamma)$ having at least three $G$-alternating cycles. Then the graph $\mathrm{Alt}_{G}(\Gamma)$ is a regular graph of valence $2\mathop{{\rm rad}}_{G}(\Gamma)/\mathop{{\rm att}}_{G}(\Gamma)$ and the induced action of $G$ on $\mathrm{Alt}_{G}(\Gamma)$ is vertex- and edge-transitive. Moreover, this action is arc-transitive if and only if $\mathop{{\rm rad}}_{G}(\Gamma)$ does not divide $\mathop{{\rm att}}_{G}(\Gamma)$.

Proof.

To simplify notation, denote $r=\mathop{{\rm rad}}_{G}(\Gamma)$ and $a=\mathop{{\rm att}}_{G}(\Gamma)$. Since each vertex of $\Gamma$ lies on exactly two $G$-alternating cycles and the intersection of any two non-disjoint $G$-alternating cycles is of size $a$ it is clear that each $G$-alternating cycle is adjacent to $\ell=2r/a$ other $G$-alternating cycles in $\mathrm{Alt}_{G}(\Gamma)$. Moreover, since $G$ acts edge-transitively on $\Gamma$ and each edge of $\Gamma$ is contained in a unique $G$-alternating cycle, the induced action of $G$ on $\mathrm{Alt}_{G}(\Gamma)$ is vertex-transitive. That this action is also edge-transitive follows from the fact that $G$ acts vertex-transitively on $\Gamma$ and that the edges of $\mathrm{Alt}_{G}(\Gamma)$ correspond to $G$-attachment sets of $\Gamma$.

For the rest of the proof fix one of the two paired orientations of $\Gamma$ given by the action of $G$, let $C=(v_{0},v_{1},\ldots,v_{2r-1})$ be a $G$-alternating cycle such that $v_{0}\to v_{1}$ and let $C^{\prime}$ be the other $G$-alternating cycle containing $v_{0}$, so that $C\cap C^{\prime}=\{v_{i\ell}\colon 0\leq i<a\}$. Since every other vertex of $C$ is the tail of the two edges of $C$ incident to it, the vertex $v_{\ell}$ is the tail of the two edges of $C$ incident to it if and only if $\ell$ is even (in which case each $v_{i\ell}$ has this property).

Now, if $\ell$ is odd, then each element of $G$, mapping $v_{0}$ to $v_{\ell}$ necessarily interchanges $C$ and $C^{\prime}$, proving that in this case the induced action of $G$ on $\mathrm{Alt}_{G}(\Gamma)$ is in fact arc-transitive. We remark that this also follows from the fact, first observed by Tutte [19], that a vertex- and edge-transitive group of automorphisms of a graph of odd valence is necessarily arc-transitive. To complete the proof we thus only need to show that the induced action of $G$ on $\mathrm{Alt}_{G}(\Gamma)$ is not arc-transitive when $\ell$ is even. Recall that in this case each vertex $v_{i\ell}\in C\cap C^{\prime}$ is the tail of the two edges of $C$ incident to it. Therefore, since any element of $G$, mapping the pair $\{C,C^{\prime}\}$ to itself of course preserves the intersection $C\cap C^{\prime}$ it is clear that any such element fixes each of $C$ and $C^{\prime}$ setwise, and so no element of $G$ can interchange $C$ and $C^{\prime}$. This proves that the induced action of $G$ on $\mathrm{Alt}_{G}(\Gamma)$ is half-arc-transitive. ∎

2.2. The dart graph and its relation to $\mathrm{Alt}_{G}(\Gamma)$

The dart graph of a cubic graph was investigated in [4] (we remark that this construction can also be viewed as a special kind of the arc graph construction from [3]). Of course the dart graph construction can be applied to arbitrary graphs but here, as in [4], we are only interested in dart graphs of cubic graphs. We first recall the definition. Let $\Lambda$ be a cubic graph. Then its dart graph $\mathop{{\rm Dart}}(\Lambda)$ is the graph whose vertex set consists of all the arcs (called darts in [4]) of $\Lambda$ with $(u,v)$ adjacent to $(u^{\prime},v^{\prime})$ if and only if either $u^{\prime}=v$ but $u\neq v^{\prime}$, or $u=v^{\prime}$ but $u^{\prime}\neq v$. In other words, the edges of $\mathop{{\rm Dart}}(\Lambda)$ correspond to the $2$-arcs of $\Lambda$. Note that this enables a natural orientation of the edges of $\mathop{{\rm Dart}}(\Lambda)$ where the edge $(u,v)(v,w)$ is oriented from $(u,v)$ to $(v,w)$.

Clearly, $\mathrm{Aut}(\Lambda)$ can be viewed as a subgroup of $\mathrm{Aut}(\mathop{{\rm Dart}}(\Lambda))$ preserving the natural orientation. Furthermore, the permutation $\tau$ of $V(\mathop{{\rm Dart}}(\Lambda))$, exchanging each $(u,v)$ with $(v,u)$, is an orientation reversing automorphism of $\mathop{{\rm Dart}}(\Lambda)$.

We now establish the correspondence between the $2$-arc-transitive cubic graphs and the tetravalent graphs admitting a half-arc-transitive group of automorphisms with the corresponding radius $3$ and attachment number $2$. We do this in two steps.

Proposition 5**.**

Let $\Lambda$ be a connected cubic graph admitting a $2$-arc-transitive group of automorphisms $G$ and let $\Gamma=\mathop{{\rm Dart}}(\Lambda)$. Then $\Gamma$ is a tetravalent $G$-half-arc-transitive graph such that $\mathop{{\rm rad}}_{G}(\Gamma)=3$ and $\mathop{{\rm att}}_{G}(\Gamma)=2$ with $\mathrm{Alt}_{G}(\Gamma)\cong\Lambda$. Moreover, the natural orientation of $\Gamma$, viewed as $\mathop{{\rm Dart}}(\Lambda)$, coincides with one of the two paired orientations induced by the action of $G$.

Proof.

That the natural action of $G$ on $\Gamma$ is half-arc-transitive can easily be verified (see also [4]). Now, fix an edge $(u,v)(v,w)$ of $\Gamma$ and choose the $G$-induced orientation of $\Gamma$ in such a way that $(u,v)\to(v,w)$. Since $G$ is $2$-arc-transitive on $\Lambda$, the other edge of $\Gamma$, for which $(u,v)$ is its tail, is $(u,v)(v,w^{\prime})$, where $w^{\prime}$ is the remaining neighbour of $v$ in $\Lambda$ (other than $u$ and $w$). It is now clear that for each pair of adjacent vertices $(x,y)$ and $(y,z)$ of $\Gamma$ the corresponding edge is oriented from $(x,y)$ to $(y,z)$, and so the chosen $G$-induced orientation of $\Gamma$ is the natural orientation of $\mathop{{\rm Dart}}(\Lambda)$.

Finally, let $v$ be a vertex of $\Lambda$ and let $u,u^{\prime},u^{\prime\prime}$ be its three neighbours. The $G$-alternating cycle of $\Gamma$ containing the edge $(u,v)(v,u^{\prime})$ is then clearly $C_{v}=((u,v),(v,u^{\prime}),(u^{\prime\prime},v),(v,u),(u^{\prime},v),(v,u^{\prime\prime}))$, implying that $\mathop{{\rm rad}}_{G}(\Gamma)=3$. This also shows that the $G$-alternating cycles of $\Gamma$ naturally correspond to vertices of $\Lambda$. Since the three $G$-alternating cycles of $\Gamma$ that have a nonempty intersection with $C_{v}$ are the ones corresponding to the vertices $u$, $u^{\prime}$ and $u^{\prime\prime}$, this correspondence in fact shows that $\mathrm{Alt}_{G}(\Gamma)$ and $\Lambda$ are isomorphic and that $\mathop{{\rm att}}_{G}(\Gamma)=2$. ∎

Proposition 6**.**

Let $\Gamma$ be a connected tetravalent $G$-half-arc-transitive graph for some $G\leq\mathrm{Aut}(\Gamma)$ with $\mathop{{\rm rad}}_{G}(\Gamma)=3$ and $\mathop{{\rm att}}_{G}(\Gamma)=2$, and let $\Lambda=\mathrm{Alt}_{G}(\Gamma)$. Then the group $G$ induces a $2$-arc-transitive action on $\Lambda$ and $\mathop{{\rm Dart}}(\Lambda)\cong\Gamma$. In fact, an isomorphism $\Psi\colon\mathop{{\rm Dart}}(\Lambda)\to\Gamma$ exists which maps the natural orientation of $\mathop{{\rm Dart}}(\Lambda)$ to a $G$-induced orientation of $\Gamma$.

Proof.

By Proposition 4 the graph $\Lambda$ is cubic and the induced action of $G$ on it is arc-transitive. Since $\mathop{{\rm rad}}_{G}(\Gamma)=3$ and $\mathop{{\rm att}}_{G}(\Gamma)=2$ it is easy to see that $\Gamma$ and $\mathop{{\rm Dart}}(\Lambda)$ are of the same order. Furthermore, let $C=(v_{0},v_{1},\ldots,v_{5})$ be a $G$-alternating cycle of $\Gamma$ and $C^{\prime},C^{\prime\prime},C^{\prime\prime\prime}$ be the other $G$-alternating cycles of $\Gamma$ containing $v_{0},v_{1}$ and $v_{5}$, respectively. Then $C\cap C^{\prime}=\{v_{0},v_{3}\}$, $C\cap C^{\prime\prime}=\{v_{1},v_{4}\}$ and $C\cap C^{\prime\prime\prime}=\{v_{2},v_{5}\}$. It is thus clear that any element of $G$, fixing $v_{0}$ and mapping $v_{1}$ to $v_{5}$ (which exists since $C$ is $G$-alternating and $G$ is edge-transitive on $\Gamma$), fixes both $C$ and $C^{\prime}$ but maps $C^{\prime\prime}$ to $C^{\prime\prime\prime}$. Therefore, the induced action of $G$ on $\Lambda$ is $2$-arc-transitive.

To complete the proof we exhibit a particular isomorphism $\Psi\colon\mathop{{\rm Dart}}(\Lambda)\to\Gamma$. Fix an orientation of the edges of $\Gamma$, induced by the action of $G$, and let $C$ and $C^{\prime}$ be two $G$-alternating cycles of $\Gamma$ with a nonempty intersection. Then $(C,C^{\prime})$ and $(C^{\prime},C)$ are vertices of $\mathop{{\rm Dart}}(\Lambda)$. Let $C\cap C^{\prime}=\{u,u^{\prime}\}$ and observe that precisely one of $u$ and $u^{\prime}$ is the head of both of the edges of $C$ incident to it. Without loss of generality assume it is $u$. Then of course $u^{\prime}$ is the head of both of the edges of $C^{\prime}$ incident to it. We then set $\Psi((C,C^{\prime}))=u$ and $\Psi((C^{\prime},C))=u^{\prime}$. Therefore, for non-disjoint $G$-alternating cycles $C$ and $C^{\prime}$ of $\Gamma$ we map $(C,C^{\prime})$ to the unique vertex in $C\cap C^{\prime}$ which is the head of both of the edges of $C$ incident to it. Since each pair of non-disjoint $G$-alternating cycles meets in precisely two vertices and each vertex of $\Gamma$ belongs to two $G$-alternating cycles of $\Gamma$, this mapping is injective and thus also bijective. We now only need to show that it preserves adjacency and maps the natural orientation of $\mathop{{\rm Dart}}(\Lambda)$ to the chosen $G$-induced orientation of $\Gamma$. To this end let $C$, $C^{\prime}$ and $C^{\prime\prime}$ be three $G$-alternating cycles of $\Gamma$ such that $C$ has a nonempty intersection with both $C^{\prime}$ and $C^{\prime\prime}$. Recall that then the edge $(C^{\prime},C)(C,C^{\prime\prime})$ is oriented from $(C^{\prime},C)$ to $(C,C^{\prime\prime})$ in the natural orientation of $\mathop{{\rm Dart}}(\Lambda)$. Denote $C=(v_{0},v_{1},\ldots,v_{5})$ and without loss of generality assume $C\cap C^{\prime}=\{v_{0},v_{3}\}$ and $C\cap C^{\prime\prime}=\{v_{1},v_{4}\}$.

Suppose first that $v_{0}\to v_{1}$. Then $v_{0}$ is the head of both of the edges of $C^{\prime}$ incident to it, and so $\Psi((C^{\prime},C))=v_{0}$. Similarly, $v_{1}$ is the head of both of the edges of $C$ incident to it, and so $\Psi((C,C^{\prime\prime}))=v_{1}$. If on the other hand $v_{1}\to v_{0}$, then $\Psi((C^{\prime},C))=v_{3}$ and $\Psi((C,C^{\prime\prime}))=v_{4}$. In both cases, $\Psi$ maps the oriented edge $(C^{\prime},C)(C,C^{\prime\prime})$ to an oriented edge of $\Gamma$, proving that it is an isomorphism of graphs, mapping the the natural orientation of $\mathop{{\rm Dart}}(\Lambda)$ to the chosen $G$-induced orientation of $\Gamma$. ∎

Theorem  3 now follows directly from Propositions 5 and 6.

3. Partial answer to Question 1 and proof of Theorem 2

In this section we prove Theorem 2 giving a partial answer to Question 1. We first prove an auxiliary result.

Proposition 7**.**

Let $\Gamma$ be a tetravalent $G$-half-arc-transitive graph with $\mathop{{\rm att}}_{G}(\Gamma)$ even. Then for each vertex $v$ of $\Gamma$ and the two $G$-alternating cycles $C$ and $C^{\prime}$, containing $v$, the antipodal vertex of $v$ on $C$ coincides with the antipodal vertex of $v$ on $C^{\prime}$. Moreover, the involution $\tau$ interchanging each pair of antipodal vertices on all $G$-alternating cycles of $\Gamma$ is an automorphism of $\Gamma$ centralising $G$.

Proof.

Denote $r=\mathop{{\rm rad}}_{G}(\Gamma)$ and $a=\mathop{{\rm att}}_{G}(\Gamma)$. Let $v$ be a vertex of $\Gamma$ and let $C$ and $C^{\prime}$ be the two $G$-alternating cycles of $\Gamma$ containing $v$. Denote $C=(v_{0},v_{1},\ldots,v_{2r-1})$ with $v=v_{0}$. Recall that then $C\cap C^{\prime}=\{v_{i\ell}\colon 0\leq i<a\}$, where $\ell=2r/a$. Since $a$ is even $v_{r}\in C\cap C^{\prime}$. Now, take any element $g\in G_{v}$ interchanging $v_{1}$ with $v_{2r-1}$ as well as the other two neighbours of $v$ (which are of course neighbours of $v$ on $C^{\prime}$). Then $g$ reflects both $C$ and $C^{\prime}$ with respect to $v$. Since $v_{r}$ is antipodal to $v$ on $C$, it must be fixed by $g$, but since $v_{r}$ is also contained in $C^{\prime}$, this implies that it is in fact also the antipodal vertex of $v$ on $C^{\prime}$. This shows that for each $G$-alternating cycle $C$ and each vertex $v$ of $C$ the vertex $v$ and its antipodal counterpart on $C$ both belong to the same pair of $G$-alternating cycles (this implies that the $G$-transversals, as they were defined in [8], are of length $2$) and are also antipodal on the other $G$-alternating cycle containing them.

It is now clear that $\tau$ is a well defined involution on the vertex set of $\Gamma$. Since the antipodal vertex of a neighbor $v_{1}$ of $v=v_{0}$ on $C$ is the neighbor $v_{r+1}$ of the antipodal vertex $v_{r}$, it is clear that $\tau$ is in fact an automorphism of $\Gamma$. Since any element of $G$ maps $G$-alternating cycles to $G$-alternating cycles it is clear that $\tau$ centralises $G$. ∎

We are now ready to prove Theorem 2. Let $\Gamma$ be a tetravalent half-arc-transitive graph. Denote $r=\mathop{{\rm rad}}(\Gamma)$ and $a=\mathop{{\rm att}}(\Gamma)$, and assume $r$ is odd. Recall that $a$ divides $2r$. We thus only need to prove that $a$ is odd. Suppose to the contrary that $a$ is even, and so by assumption $a\equiv 2\pmod{4}$. Then the graph $\Gamma$ admits the automorphism $\tau$ from Proposition 7. Now, fix one of the two paired orientations of the edges induced by the action of $\mathrm{Aut}(\Gamma)$ and let $C=(v_{0},v_{1},\ldots,v_{2r-1})$ be an alternating cycle of $\Gamma$ with $v_{0}$ being the tail of the edge $v_{0}v_{1}$. Since $v_{0}^{\tau}=v_{r}$ and $v_{1}^{\tau}=v_{r+1}$ it follows that $v_{r}$ is the tail of the edge $v_{r}v_{r+1}$. But since $r$ is odd this contradicts the fact that every other vertex of $C$ is the tail of the two edges of $C$ incident to it. Thus $a$ is odd, as claimed.

To prove the second part of the theorem assume that $a$ is not divisible by $4$. If $r$ is even then the fact that $a$ divides $2r$ implies that $a$ divides $r$ as well. If however $r$ is odd, we can apply the first part of the theorem. This completes the proof.

4. An infinite family of non-sectional split covers

As announced in the introduction, tetravalent $G$-half-arc-transitive graphs $\Gamma$ with $\mathop{{\rm rad}}_{G}(\Gamma)=3$ and $\mathop{{\rm att}}_{G}(\Gamma)=2$ yield surprising examples of the elusive non-sectional split covers. In this section, we present this connection in some detail.

Theorem 8**.**

Let $\Gamma$ be a connected non-bipartite $G$-half-arc-transitive graph $\Gamma$ of order greater than $12$ with $\mathop{{\rm rad}}_{G}(\Gamma)=3$ and $\mathop{{\rm att}}_{G}(\Gamma)=2$. Then there exists a $2$-fold covering projection $\wp\colon\Gamma\to\Gamma^{\prime}$ and an arc-transitive group $H\leq\mathrm{Aut}(\Gamma^{\prime})$ which lifts along $\wp$ in such a way that $\Gamma$ is a non-sectional $H$-split cover of $\Gamma^{\prime}$.

Proof.

Since $\mathop{{\rm att}}_{G}(\Gamma)=2$, each $G$-attachment set consists of a pair of antipodal vertices on a $G$-alternating cycle of $\Gamma$. Let $\mathcal{B}$ be the set of all $G$-attachment sets in $\Gamma$. By Proposition 7, there exists an automorphism $\tau$ of $\Gamma$ centralising $G$, which interchanges the two vertices in each element of $\mathcal{B}$. Let $\tilde{G}=\langle G,\tau\rangle$ and note that $\tilde{G}$ acts transitively on the arcs of $\Gamma$. Since $\tau$ is an involution centralising $G$ not contained in $G$, we see that $\tilde{G}=G\times\langle\tau\rangle$.

Let $\Gamma^{\prime}$ be the quotient graph with respect to the group $\langle\tau\rangle$, that is, the graph whose vertices are the orbits of $\langle\tau\rangle$ and with two such orbits adjacent whenever they are joined by an edge in $\Gamma$. Since $\tilde{G}$ is arc-transitive and $\langle\tau\rangle$ is normal in $\tilde{G}$, each $\langle\tau\rangle$-orbit is an independent set. Moreover, if two $\langle\tau\rangle$-orbits $B$ and $C$ are adjacent in $\Gamma^{\prime}$, then the induced subgraph $\Gamma[B\cup C]$ is clearly vertex- and arc-transitive and is thus either $K_{2,2}$ or $2K_{2}$. In the former case, it is easy to see that $\Gamma$ is isomorphic to the lexicographic product of a cycle with the edge-less graph on two vertices. Since $\mathop{{\rm rad}}_{G}(\Gamma)=3$ and the orbits of $\langle\tau\rangle$ coincide with the elements of $\mathcal{B}$, this implies that $\Gamma$ has only $6$ vertices, contradicting our assumption on the order of $\Gamma$. This contradiction implies that $\Gamma[B\cup C]\cong 2K_{2}$ for any pair of adjacent $\langle\tau\rangle$-orbits $B$ and $C$, and hence the quotient projection $\wp\colon\Gamma\to\Gamma^{\prime}$ is a $2$-fold covering projection with $\langle\tau\rangle$ being its group of covering transformations.

Since $\tau$ normalises $G$, the group $\tilde{G}$ projects along $\wp$ and the quotient group $H=\tilde{G}/\langle\tau\rangle$ acts faithfully as an arc-transitive group of automorphisms on $\Gamma^{\prime}$. In particular, since the group of covering projection $\langle\tau\rangle$ has a complement $G$ in $\tilde{G}$, the covering projection $\wp$ is $H$-split.

By [2, Proposition 3.3], if $\wp$ had a sectional complement with respect to $H$, then $\Gamma$ would be a canonical double cover of $\Gamma^{\prime}$, contradicting the assumption that $\Gamma$ is not bipartite. ∎

Remark. In [4, Proposition 9] it was shown that a cubic graph $\Lambda$ is bipartite if and only if $\mathop{{\rm Dart}}(\Lambda)$ is bipartite. Since there exist infinitely many connected non-bipartite cubic $2$-arc-transitive graphs, Theorem 3 thus implies that there are indeed infinitely many connected non-bipartite $G$-half-arc-transitive graphs $\Gamma$ with $\mathop{{\rm rad}}_{G}(\Gamma)=3$ and $\mathop{{\rm att}}_{G}(\Gamma)=2$. In view of Theorem 8, these yield infinitely many non-sectional split covers, as announced in the introduction. Furthermore, note that the $G$-alternating $6$-cycles in the graph $\Gamma$ appearing in the proof of the above theorem project by $\wp$ to cycles of length $3$, implying that $\Gamma^{\prime}$ is a tetravalent arc-transitive graph of girth $3$. Since it is assumed that the order of $\Gamma$ is larger than $12$ (and thus the order of $\Gamma^{\prime}$ is larger than $6$), we may now use [17, Theorem 5.1] to conclude that $\Gamma^{\prime}$ is isomorphic to the line graph of a $2$-arc-transitive cubic graph.

Acknowledgment. The first author was supported in part by Slovenian Research Agency, program P1-0294. The second author was supported in part by Slovenian Research Agency, program P1-0285 and projects N1-0038, J1-6720 and J1-7051.