NOWHERE-ZERO $3$-FLOWS IN TWO FAMILIES OF VERTEX-TRANSITIVE GRAPHS

Abstract

Let $\Gamma $ be a graph of valency at least four whose automorphism group contains a minimally vertex-transitive subgroup G. It is proved that $\Gamma $ admits a nowhere-zero $3$ -flow if one of the following two conditions holds: (i) $\Gamma $ is of order twice an odd number and G contains a central involution; (ii) G is a direct product of a $2$ -subgroup and a subgroup of odd order.

1 Introduction

Graphs considered in this paper are finite, undirected, loopless, but allowed to have multiple edges. Let $\Gamma $ be a graph. As usual, we use $V(\Gamma )$ and $E(\Gamma )$ to denote the vertex set and edge set of $\Gamma $ , respectively. An orientation $\mathcal {D}$ of $\Gamma $ is an assignment of one of the two possible orientations for every $e\in E(\Gamma )$ . Let $\varphi $ be a mapping from $E(\Gamma )$ to the set of integers and k a positive integer. For every $v\in V(G)$ , we use $\varphi ^{+}(v)$ to denote the sum of values $\varphi (e)$ of edges e with orientation originating from v and $\varphi ^{-}(v)$ the sum of values $\varphi (e)$ of edges e with orientation pointing to v. If $-k<\varphi (e)<k$ for every $e\in E(\Gamma )$ and $\varphi ^{+}(v)=\varphi ^{-}(v)$ for every $v\in V(\Gamma )$ , then we call the ordered pair $(\mathcal {D}, \varphi )$ a k-flow of $\Gamma $ . If further $\varphi (e)\neq 0$ for every $e\in E(\Gamma )$ , then we call $(\mathcal {D}, \varphi )$ a nowhere-zero k-flow of $\Gamma $ . For convenience, we use $\mathcal {NZ}_k$ to denote the family of graphs which admit a nowhere-zero k-flow.

Tutte proposed three conjectures in the middle of the last century on integer flows which are still unsolved, namely the $5$ -flow, $4$ -flow and $3$ -flow conjectures. The $3$ -flow conjecture (see, for example, [16, Conjecture 1.1.8]) is stated as follows: every $4$ -edge-connected graph is contained in $\mathcal {NZ}_3$ . By the equivalent version of the $3$ -flow conjecture given by Kochol [6], it suffices to prove this conjecture for $5$ -edge-connected graphs. However, it was conjectured by Jaeger [5] that every k-edge-connected graph is contained in $\mathcal {NZ}_3$ for some given positive integer k. This so called weak $3$ -flow conjecture was solved by Thomassen [14] who proved that the statement holds for an $8$ -edge-connected graph. Lovász et al. [8] improved this breakthrough by proving that the statement of the weak $3$ -flow conjecture is true when $k=6$ . However, the $3$ -flow conjecture remains wide open for $5$ -edge-connected graphs. In this situation, it is natural to attempt to verify the conjecture for interesting families of graphs, for example, vertex-transitive graphs.

A graph $\Gamma $ is vertex-transitive if its automorphism group $\mathrm {Aut}(\Gamma )$ acts transitively on $V(\Gamma )$ . A subgroup G of $\mathrm {Aut}(\Gamma )$ is said to be minimally vertex-transitive if G is transitive on $V(\Gamma )$ , but any proper subgroup of G is intransitive on $V(\Gamma )$ . In particular, if G acts regularly (transitively and every nontrivial element fixes no vertex) on $V(\Gamma )$ , then $\Gamma $ is called a Cayley graph on G. A graph is said to be k-regular (or regular for short) if each of its vertices has valency k where k is a positive integer. It is obvious that every vertex-transitive graph is regular. In [9], it was proved that the edge connectivity of a connected vertex-transitive simple graph is equal to its valency. Thus, the $3$ -flow conjecture for vertex-transitive graphs asserts that every vertex-transitive simple graph of valency at least four is contained in $\mathcal {NZ}_3$ . In this direction, the $3$ -flow conjecture was verified for Cayley graphs on abelian groups [11], nilpotent groups [10], dihedral groups [15], generalised dihedral groups [7], generalised quaternion groups [7], generalised dicyclic groups [1], groups of order $pq^2$ (p and q are two primes) (J. Zhang and Z. Zhang, ‘Nowhere-zero 3-flows in Cayley graphs of order $pq^2$ ’, submitted for publication) and two families of supersolvable groups [17]. Very recently, the first author and Zhou [18] proved that a graph of valency at least four is contained in $\mathcal {NZ}_3$ if its automorphism group has a vertex-transitive nilpotent subgroup.

In [10], to study nowhere-zero $3$ -flows in Cayley graphs, Nánásiová and Škoviera introduced the method of decomposing a graph into a union of closed ladders. They proved that a Cayley graph of valency at least four on a group G is contained in $\mathcal {NZ}_3$ if its connected set contains a central involution of G. They also proved that every Cayley graph of valency at least four on a group which is a direct product of a $2$ -subgroup and a subgroup of odd order is contained in $\mathcal {NZ}_3$ . In this paper, we attempt to generalise the above two results to vertex-transitive graphs. We obtain the following two theorems.

Theorem 1.1. Let $\Gamma $ be a vertex-transitive graph of order twice an odd number and valency at least $4$ . Let G be a minimally vertex-transitive subgroup of the automorphism group of $\Gamma $ . If G contains a central involution, then $\Gamma \in \mathcal {NZ}_3$ .

Theorem 1.2. Let $\Gamma $ be a graph of valency at least four. If there exists a subgroup of $\mathrm {Aut}(\Gamma )$ which acts transitively on $V(\Gamma )$ and is a direct product of a $2$ -subgroup and a subgroup of odd order, then $\Gamma \in \mathcal {NZ}_3$ .

The proof of Theorem 1.2 relies on Theorem 1.1 and the main result of [18].

2 Preparations

Let $\Gamma _1$ and $\Gamma _2$ be two graphs. The Cartesian product $\Gamma :=\Gamma _1\square \Gamma _2$ of $\Gamma _1$ and $\Gamma _2$ is the graph defined as follows:

• • $V(\Gamma )=V(\Gamma _1)\times V(\Gamma _2)$ ;

• • $E(\Gamma )=V(\Gamma _1)\times E(\Gamma _2)\cup E(\Gamma _1)\times V(\Gamma _2)$ ;

• • each $(u_{1},e_2)\in V(\Gamma _1)\times E(\Gamma _2)$ is an edge with ends $(u_{1},u_2)$ and $(u_{1},v_2)$ , where $e_2$ is an edge in $E(\Gamma _2)$ with ends $u_2$ and $v_2$ ;

• • each $(e_{1},u_2)\in E(\Gamma _1)\times V(\Gamma _2)$ is an edge with ends $(u_{1},u_2)$ and $(v_{1},u_2)$ , where $e_1$ is an edge in $E(\Gamma _1)$ with ends $u_1$ and $v_1$ .

Let $C_n$ be the cycle of length n which has vertex set $\{1,\ldots ,n\}$ and edge set $\{\{1,2\},\ldots ,\{n-1,n\},\{n,1\}\}$ . Let $K_2$ be the complete graph of order two with vertex set $\{0,1\}$ . The graph $CL_n:=C_n \Box K_2$ is called a circular ladder. The Möbius ladder $M_n$ is a graph obtained from $CL_n$ by replacing the edges $\{(1,0),(n,0)\}$ and $\{(1,1),(n,1)\}$ with $\{(1,0),(n,1)\}$ and $\{(1,1),(n,0)\}$ , respectively (see Figure 1). A graph is called a closed ladder if it is a circular ladder or a Möbius ladder. In a closed ladder $\Gamma $ , every edge of the form $\{(i,0),(i,1)\}$ is called a rung of $\Gamma $ . We use $R(\Gamma )$ to denote the set of all rungs of $\Gamma $ .

Figure 1 $CL_n$ and $M_n$ .

In [10], the method of decomposing a graph into a union of closed ladders was introduced to study nowhere-zero $3$ -flows in Cayley graphs. The following lemma, given in [7], is derived from the proof of [10, Theorem 3.3].

Lemma 2.1. Let $\Gamma :=\bigcup _{i=1}^{s}\Theta _i$ be a connected graph where every subgraph $\Theta _i$ is a closed ladder. If $E(\Theta _i)\cap E(\Theta _j)=R(\Theta _i)\cap R(\Theta _j)$ for any pair of distinct $i,j\in \{1,\ldots ,s\}$ and each edge of $\bigcup _{i=1}^{s}R(\Theta _i)$ is contained in at least two closed ladders in $\{\Theta _1,\ldots ,\Theta _s\}$ , then $\Gamma $ is contained in $\mathcal {NZ}_3$ .

A graph is said to be even if each of its vertices is of even valency. It is well known [2, Theorem 21.4] that every even graph is contained in $\mathcal {NZ}_k$ for all integers $k\geq 2$ . Let $\Gamma $ be a graph and E a subset of $E(\Gamma )$ . We use $\Gamma -E$ to denote the subgraph of $\Gamma $ with vertex set $V(\Gamma )$ and edge set $E(\Gamma )-E$ . A subgraph $\Gamma '$ of $\Gamma $ is called a parity subgraph of $\Gamma $ if $\Gamma -E(\Gamma ')$ is even. It is also well known [2, Theorem 21.5] that a cubic bipartite graph is contained in $\mathcal {NZ}_3$ . All that leads to the following obvious lemma.

Lemma 2.2. Let $\Gamma $ be a graph and $\Gamma '$ a parity subgraph of $\Gamma $ . If $\Gamma '\in \mathcal {NZ}_3$ , then $\Gamma \in \mathcal {NZ}_3$ . In particular, if every vertex of $\Gamma $ is of odd valency and $\Gamma '$ is a spanning cubic bipartite subgraph of $\Gamma $ , then $\Gamma \in \mathcal {NZ}_3$ .

In [4], it was proved that the Cartesian product of two nontrivial connected bipartite graphs is contained in $\mathcal {NZ}_3$ . This result was generalised in [13] by proving that the Cartesian product of every pair of graphs is contained in $\mathcal {NZ}_3$ except when one factor has a cut edge and every block of another factor is a circuit of odd length. By using Lemma 2.1, we prove the following lemma.

Lemma 2.3. Let $\Gamma _1$ be an even graph with minimum valency at least four and $\Gamma _2$ be an arbitrary graph. Then $\Gamma _1\square \Gamma _2$ is contained in $\mathcal {NZ}_3$ .

Proof. Let $\Gamma _{11},\ldots ,\Gamma _{1m}$ be all the connected components of $\Gamma _{1}$ . Then $\Gamma _1\square \Gamma _2$ is an edge-disjoint union of $\Gamma _{11}\square \Gamma _2,\ldots ,\Gamma _{1m}\square \Gamma _2$ . Therefore, $\Gamma _1\square \Gamma _2\in \mathcal {NZ}_3$ if and only if $\Gamma _{1i}\square \Gamma _2\in \mathcal {NZ}_3$ for every $1\leq i\leq m$ . Since $\Gamma _{1}$ is an even graph with minimum valency at least four, $\Gamma _{1i}$ is an even graph with minimum valency at least four for every $1\leq i\leq m$ . Therefore, we assume that $\Gamma _1$ is connected (for otherwise, we consider its components). By Veblen’s theorem [2, Theorem 2.7], every even graph is an edge disjoint union of cycles. Therefore, there is a family $\mathcal {F}_1$ of edge disjoint cycles of $\Gamma _1$ such that $\bigcup _{\Sigma \in \mathcal {F}_1}\Sigma =\Gamma _1$ . Let $\mathcal {F}_2$ be the decomposition of $\Gamma _2$ such that every member of $\mathcal {F}_2$ is either a complete graph of order two or a trivial graph with just one isolated vertex in $\Gamma _2$ (note that every graph has such a decomposition).

Consider an arbitrary member $\Lambda \in \mathcal {F}_2$ . If $\Lambda $ is a trivial graph, then $\Gamma _1\square \Lambda $ is isomorphic to $\Gamma _1$ and therefore an even graph. Since every even graph is contained in $\mathcal {NZ}_3$ , we have $\Gamma _1\square \Lambda \in \mathcal {NZ}_3$ . Now consider the case that $\Lambda $ is the complete graph of order two. Set $\mathcal {F}_1=\{\Sigma _{1},\ldots ,\Sigma _{s}\}$ and $\Theta _i=\Sigma _{i}\square \Lambda $ for every $\Sigma _{i}\in \mathcal {F}_1$ . Then $\mathcal {F}:=\{\Theta _{1},\ldots ,\Theta _{s}\}$ is a family of circular ladders. It is obvious that $\Gamma _1\square \Lambda =\bigcup _{i=1}^{s}\Theta _i$ . Moreover, $\Gamma _1\square \Lambda $ is connected as both $\Gamma _1$ and $\Lambda $ are connected. Let $1\leq i<j\leq s$ . Since $\Sigma _{i}$ and $\Sigma _{j}$ have no common edge, $E(\Theta _i)\cap E(\Theta _j)=R(\Theta _i)\cap R(\Theta _j)$ . Since the minimum valency of $\Gamma _1$ is at least four, every vertex of $\Gamma _1$ is contained in at least two cycles in $\mathcal {F}_1$ . It follows that each edge of $\bigcup _{i=1}^{s}R(\Theta _i)$ is contained in at least two members of $\mathcal {F}$ . By Lemma 2.1, $\Gamma _1\square \Lambda \in \mathcal {NZ}_3$ .

Set $\mathcal {F}_2=\{\Lambda _1,\ldots ,\Lambda _{t}\}$ . By the above discussion, $\Gamma _1\square \Lambda _{i}\in \mathcal {NZ}_3$ for every $\Lambda _{i}\in \mathcal {F}_2$ . Since $\Gamma _{1}\square \Gamma _2$ is the edge disjoint union of $\Gamma _1\square \Lambda _{1},\ldots ,\Gamma _1\square \Lambda _{t}$ , we have $\Gamma _{1}\square \Gamma _2\in \mathcal {NZ}_3$ .

The following lemma is extracted from [18, Lemma 4.8].

Lemma 2.4. Let $\Gamma $ be a graph of valency five whose automorphism group contains a vertex-transitive subgroup G having a central involution z. Suppose that $\Gamma $ has a perfect matching M of which every edge is of the form $\{v,z(v)\}$ , $v\in V(\Gamma )$ . If $\Gamma $ is an edge-disjoint union of M, $\Gamma _{1}$ and $\Gamma _{2}$ , where $\Gamma _{1}$ and $\Gamma _{2}$ are both spanning $2$ -regular subgraphs of $\Gamma $ preserved by G, then $\Gamma \in \mathcal {NZ}_3$ .

3 Proof of Theorems 1.1 and 1.2

Proof of Theorem 1.1

Let $\Gamma $ be a graph of order $2n$ , where n is an odd number. We assume that $\Gamma $ is of odd valency as $\Gamma \in \mathcal {NZ}_3$ if $\Gamma $ is even. Let G be a minimally vertex-transitive subgroup of the automorphism group of $\Gamma $ and z a central involution of G.

We first prove that z does not fix any vertex of $\Gamma $ . Otherwise, if $z(v)=v$ for some $v\in V(\Gamma )$ , then $z(g(v))=zg(v)=gz(v)=g(v)$ for all $g\in G$ . Since G acts transitively on $V(\Gamma )$ , it follows that z fixes all vertices of $\Gamma $ . This contradicts the fact that z is not the identity automorphism of $\Gamma $ .

Since z does not fix any vertex of $\Gamma $ and $\vert V(\Gamma )\vert =2n$ , we conclude that z is a permutation factorising into n disjoint transpositions. Therefore, z is an odd permutation on $V(\Gamma )$ as n is an odd number. Let H be a subset of G consisting of all even permutations of G. Then $z\notin H$ and H is a normal subgroup of G of index $2$ . Since both $\langle z\rangle $ and H are normal in G and $\langle z\rangle \cap H={1}$ , we get $G=\langle z\rangle \times H$ .

Since G is minimally transitive on $V(\Gamma )$ , we deduce that H is intransitive on $V(\Gamma )$ . Let u be an arbitrary vertex of $\Gamma $ . Then $\vert G:G_u\vert =|V(\Gamma )|=2n$ and $|H:H_u|$ is a nontrivial divisor of $2n$ . Since $|H:H_u|=(1/2)|G:H_u|\geq (1/2)|G:G_u|=n$ , it follows that $|H:H_u|=n$ . Therefore, the action of H on $V(\Gamma )$ has two orbits. Let U be the orbit of u under the action of H on $V(\Gamma )$ . Then $z(U)$ is the orbit of $z(u)$ under the action of H on $V(\Gamma )$ as $zh=hz$ for all $h\in H$ . Since $G=\langle z\rangle \times H$ acts transitively on $V(\Gamma )$ , we have $U\cap z(U)=\emptyset $ and $U\cup z(U)=V(\Gamma )$ . Let $\Gamma [U]$ and $\Gamma [z(U)]$ be the subgraphs of $\Gamma $ induced by U and $z(U)$ , respectively. Then $\Gamma [U]$ and $\Gamma [z(U)]$ are isomorphic and have no common edges. Since U is the orbit of u under the action of H and H preserves $\Gamma [U]$ , we conclude that $\Gamma [U]$ is a regular graph. Assume that $\Gamma [U]$ is of valency s. Since $\Gamma [U]$ and $\Gamma [z(U)]$ are isomorphic and have no common edges, $\Gamma [U]\cup \Gamma [z(U)]$ is an s-regular graph. Let $\Gamma '$ be the graph obtained from $\Gamma $ by removing all the edges of $\Gamma [U]\cup \Gamma [z(U)]$ . Then $\Gamma '$ is a regular bipartite graph with bipartition $\{U,z(U)\}$ . Assume that $\Gamma '$ is of valency t. Then $\Gamma $ is of valency $s+t$ . In particular, $s+t$ is an odd number at least five. Since $\Gamma [U]$ is an s-regular graph of odd order, s is an even number and therefore t is an odd number.

By [2, Corollary 16.5], every regular bipartite graph has a perfect matching. Therefore, $\Gamma '$ has a perfect matching. If $t\geq 3$ , then, by removing a number of perfect matchings, one can get a spanning cubic bipartite subgraph $\Gamma ''$ of $\Gamma '$ which is also a spanning cubic bipartite subgraph of $\Gamma $ . By Lemma 2.2, $\Gamma \in \mathcal {NZ}_3$ .

From now on, we assume that $t=1$ . Then there exists a permutation $\mu $ on U such that $z\mu (v)$ is the unique vertex in $z(U)$ adjacent to v for all $v\in U$ . Since $z\in \mathrm {Aut}(\Gamma )$ , we see that $z(v)$ is the unique vertex in $z(U)$ adjacent to $\mu (v)$ . Therefore, $\mu ^2(v)=v$ . It follows that $\mu $ fixes at least one vertex in U as the number n of vertices of U is odd. Without loss of generality, assume $\mu (u)=u$ . Then u is adjacent to $z(u)$ . Since H is transitive on U and $zh=hz$ for all $h\in H$ , we conclude that v is adjacent to $z(v)$ for all $v\in U$ . In other words, $\mu $ is the identity permutation. Note that $\Gamma [U]$ is an even regular graph of valency at least four. Let $\Sigma =\Gamma [U]\square K_2$ be the Cartesian product of $\Gamma [U]$ and $K_2$ , where $K_2$ is the complete graph of order two with vertex set $\{0,1\}$ . By Lemma 2.3, $\Sigma \in \mathcal {NZ}_3$ . Define a mapping $\psi $ from $V(\Gamma )$ to $V(\Sigma )$ as follows:

$$ \begin{align*} \psi(v)= \begin{cases} (v,0) & \mbox{if}~v\in U, \\ (z(v),1) & \mbox{if}~v\in z(U). \\ \end{cases} \end{align*} $$

It is straightforward to check that $\psi $ is a well-defined bijection from $V(\Gamma )$ to $V(\Sigma )$ . We further prove that $\psi $ is an isomorphism.

Let $v_1$ and $v_2$ be two vertices of $\Gamma $ . Since $\Gamma [U]$ is an induced subgraph of $\Gamma $ , every edge in $\Gamma $ joining two vertices in U is contained in $\Gamma [U]$ . Therefore, if $v_1,v_2\in U$ , then by the definition of a Cartesian product, the number of edges in $\Gamma $ joining $v_1$ and $v_2$ is equal to the number of edges in $\Sigma $ joining $(v_1,0)$ and $(v_2,0)$ . If $v_1,v_2\in z(U)$ , then $z(v_1),z(v_2)\in U$ . Since $z\in \mathrm {Aut}(\Gamma )$ , the number of edges in $\Gamma $ joining $v_1$ and $v_2$ is equal to the number joining $z(v_1)$ and $z(v_2)$ . It follows that the number of edges in $\Gamma $ joining $v_1$ and $v_2$ is equal to the number of edges in $\Sigma $ joining $(z(v_1),1)$ and $(z(v_2),1)$ . Now consider the case that one of the two vertices $v_1$ and $v_2$ is contained in U and another is contained in $z(U)$ . Without loss of generality, assume that $v_1\in U$ and $v_2\in z(U)$ . Then

$$ \begin{align*} v_1~\mbox{is adjacent to}~v_2~\mbox{in}~\Gamma & \Longleftrightarrow v_2=z(v_1)\\ & \Longleftrightarrow v_1=z(v_2)\\ & \Longleftrightarrow (v_1,0)~\mbox{is adjacent to}~(z(v_2),1)~\mbox{in}~\Sigma. \end{align*} $$

The discussion above implies that the number of edges joining $v_1$ and $v_2$ in $\Gamma $ is equal to the number of edges in $\Sigma $ joining $\psi (v_1)$ and $\psi (v_2)$ . Therefore, $\psi $ is an isomorphism from $\Gamma $ to $\Sigma $ . Since $\Sigma \in \mathcal {NZ}_3$ , we have $\Gamma \in \mathcal {NZ}_3$ .

Proof of Theorem 1.2

Let $\Gamma $ be a graph of valency at least four and G a subgroup of $\mathrm {Aut}(\Gamma )$ acting transitively on $V(\Gamma )$ and being a direct product of a $2$ -subgroup Q and a subgroup H of odd order. We assume that $\Gamma $ is of odd valency as $\Gamma \in \mathcal {NZ}_3$ whenever the valency of $\Gamma $ is even. Then $\Gamma $ is of even order and it follows that Q is nontrivial.

We proceed by induction on the order $\vert Q\vert $ of Q. By Theorem 1.1, $\Gamma \in \mathcal {NZ}_3$ if $\vert Q\vert =2$ . Now assume $\vert Q\vert>2$ . Suppose that the theorem is true for all graphs whose automorphism groups have a vertex-transitive subgroup which is a direct product of a $2$ -subgroup of order less that $\vert Q\vert $ and a subgroup of odd order.

It is well known [12, Theorem 4.2] that every $2$ -group has a nontrivial centre. Let z be an involution contained in the centre of Q. Since $G=Q\times H$ , we see that z is also contained in the centre of G. Therefore, $\langle z\rangle $ is a normal subgroup of G and z does not fix any vertex of $\Gamma $ . Set $\tilde {v}:=\{v,z(v)\}$ for every $v\in V(\Gamma )$ and $\tilde {V}:=\{\tilde {v}\mid v\in V(\Gamma )\}$ . Let $\Gamma [\tilde {v}]$ be the subgraph of $\Gamma $ induced by $\tilde {v}$ . Since G acts transitively on $V(\Gamma )$ , it follows that $\Gamma [\tilde {u}]$ and $\Gamma [\tilde {v}]$ are isomorphic for every pair of vertices $u,v\in V(\Gamma )$ . Set $\Gamma '=\bigcup _{\tilde {v}\in \tilde {V}}\Gamma [\tilde {v}]$ and $\Gamma ''=\Gamma -E(\Gamma ')$ . Then both $\Gamma '$ and $\Gamma ''$ are spanning subgraphs of $\Gamma $ preserved by G. Therefore, $\Gamma '$ and $\Gamma ''$ are both vertex-transitive. Assume that the valency of $\Gamma '$ and $\Gamma ''$ are s and t, respectively. Then $\Gamma $ is of valency $s+t$ . In particular, $s+t$ is odd.

Case 1: $s\geq 2$ . Note that every connected component of $\Gamma '$ is a graph with two vertices joined by s edges. Therefore, $\Gamma '$ is a bipartite graph. If $s\geq 3$ , then $\Gamma '$ has a spanning cubic bipartite graph which is also a spanning cubic bipartite graph of $\Gamma $ . It follows from Lemma 2.2 that $\Gamma \in \mathcal {NZ}_3$ . If $s=2$ , then t is odd. By [3, Theorem 3.51], every vertex-transitive graph of odd valency has a perfect matching. Therefore, $\Gamma ''$ has a perfect matching M. Since every connected component of $\Gamma '$ is a graph with two vertices joined by two edges, every connected component of $\Gamma '\cup M$ is a graph obtained from an even length cycle by adding a parallel edge to each edge of one of the two perfect matchings of this cycle. Therefore, $\Gamma '\cup M$ is a spanning cubic bipartite graph of $\Gamma $ and it follows from Lemma 2.2 that $\Gamma \in \mathcal {NZ}_3$ .

Case 2: $s=0$ . In this case, $\tilde {v}$ is an independent set of $\Gamma $ for every $\tilde {v}\in \tilde {V}$ . Use $\tilde {\Gamma }$ to denote the graph with vertex set $\tilde {V}$ and every pair of vertices $\tilde {u}$ and $\tilde {v}$ being joined by $\ell $ edges if and only if the subgraph $\Gamma [\tilde {u}\cup \tilde {v}]$ of $\Gamma $ induced by $\tilde {u}\cup \tilde {v}$ is $\ell $ -regular (we treat an independent set as a $0$ -regular graph). Then $\tilde {\Gamma }$ is a graph of odd valency at least five and $\Gamma $ is a cover (see [11]) of $\tilde {\Gamma }$ . Furthermore, $\mathrm {Aut}(\tilde {\Gamma })$ contains $G/\langle z\rangle $ as a subgroup acting transitively on the vertex set $\tilde {V}$ of $\tilde {\Gamma }$ . Note that $G/\langle z\rangle =Q/\langle z\rangle \times H\langle z\rangle /\langle z\rangle $ and $Q/\langle z\rangle $ is of order less than Q. By the induction hypothesis, $\tilde {\Gamma }\in \mathcal {NZ}_3$ . It is known [11, Proposition 2.3] that if a graph admits a nowhere-zero $3$ -flow, then each of its covers does too. Therefore, $\Gamma \in \mathcal {NZ}_3$ .

Case 3: $s=1$ . In this case, $\Gamma '$ is a perfect matching of $\Gamma $ . Since every group of odd order is solvable, H is a solvable group. Let $H'$ be the derived subgroup of H. Then $H'$ is a proper subgroup of H and normal in G. If H is abelian, then G ( $=Q\times H$ ) is nilpotent. By [18, Theorem 1.1], $\Gamma \in \mathcal {NZ}_3$ . Now assume that H is nonabelian. Then $H'$ is nontrivial and $G/H'=QH'/H'\times H/H'$ is nilpotent. Use $\bar {v}$ to denote the orbit of v under the action of $H'$ and $\Gamma [\bar {v}]$ the subgraph of $\Gamma $ induced by $\bar {v}$ . Then $\Gamma [\bar {v}]$ is a vertex-transitive graph of odd order as $H'$ acts transitively on $\bar {v}$ . Therefore, $\Gamma [\bar {v}]$ is a regular graph of even valency, say r. Set $\Sigma =\bigcup _{v\in V(\Gamma )}\Gamma [\bar {v}]$ . Then $\Gamma '\cup \Sigma $ is of valency $r+1$ and the automorphism group of every connected component of $\Gamma '\cup \Sigma $ contains $\langle z\rangle \times H'$ as a subgroup acting transitively on the vertex set. If $r\geq 4$ , then by Theorem 1.1, every connected component of $\Gamma '\cup \Sigma $ is contained in $\mathcal {NZ}_3$ and therefore, $\Gamma '\cup \Sigma \in \mathcal {NZ}_3$ . Since $\Gamma '\cup \Sigma $ is a parity subgraph of $\Gamma $ , it follows from Lemma 2.2 that $\Gamma \in \mathcal {NZ}_3$ . If $t-r\geq 4$ , then the subgraph $\Gamma ^{*}:=\Gamma -E(\Sigma )$ of $\Gamma $ is of odd valency at least five. Let $\overline {\Gamma ^{*}}$ be the graph with vertex set $\{\bar {v}\mid v\in V\}$ and every pair of vertices $\bar {u}$ and $\bar {v}$ being joined by $\ell $ -edges if and only if $\bar {u}\cup \bar {v}$ induces a $\ell $ -regular subgraph of $\Gamma ^{*}$ . Then $\Gamma ^{*}$ is a cover of $\overline {\Gamma ^{*}}$ . Note that $\mathrm {Aut}(\overline {\Gamma ^{*}})$ contains $QH'/H'\times H/H'$ as a subgroup acting transitively on the vertex set. Since $QH'/H'\times H/H'$ is nilpotent, it follows from [18] that $\overline {\Gamma ^{*}}\in \mathcal {NZ}_3$ . By [11, Proposition 2.3], $\Gamma ^{*}\in \mathcal {NZ}_3$ . Since $\Gamma ^{*}$ is a parity subgraph of $\Gamma $ , by Lemma 2.2, we have $\Gamma \in \mathcal {NZ}_3$ . Now we assume that both r and $t-r$ are less than four. Since $t\geq 4$ and r is even, we have $r=t-r=2$ . Then $\Sigma $ and $\Gamma ''-E(\Sigma )$ are both spanning $2$ -regular subgraphs of $\Gamma $ preserved by G. Note that $\Gamma '$ is a perfect matching of which every edge is of the form $\{v,z(v)\}$ . Note also that $\Gamma $ is an edge-disjoint union of $\Gamma '$ , $\Sigma $ and $\Gamma ''-E(\Sigma )$ . By Lemma 2.4, $\Gamma \in \mathcal {NZ}_3$ .