2. Notation and preliminaries

Throughout the paper, we use standard graph theory notation and terminology. For $k\in \mathbb {N}$ , we sometimes denote the set $\{1,2,\ldots ,k\}$ by $[k]$ . For a digraph G, we denote its vertex set by $V(G)$ and its edge set $E(G)$ , and sometimes write $|G| := |V(G)|$ and $e(G) := |E(G)|$ . For $a,b \in V(G)$ , we write $ab$ for the directed edge from a to b. We write $H \subseteq G$ to mean H is a subdigraph of G (i.e., $V(H) \subseteq V(G)$ and $E(H) \subseteq E(G)$ ). We sometimes think of $F \subseteq E(G)$ as a subdigraph of G with vertex set consisting of those vertices incident to edges in F and with edge set F. We write $G-F$ for the digraph obtained from G by deleting the edges in F. For $S \subseteq V(G)$ , we write $G[S]$ for the subdigraph of G induced by S and $G-S$ for the digraph $G[V(G) \setminus S]$ . For $A,B \subseteq V(G)$ not necessarily disjoint, we define $E_G(A,B):= \{ab \in E(G): a \in A, \; b \in B \}$ and we write $e_G(A,B) := |E_G(A,B)|$ Footnote ¹. We often drop subscripts if these are clear from context. For two digraphs $H_1$ and $H_2$ , the union $H_1 \cup H_2$ is the digraph with vertex set $V(H_1) \cup V(H_2)$ and edge set $E(H_1) \cup E(H_2)$ . We say an undirected graph G is bipartite with bipartition $A,B$ if $V(G)=A\cup B$ and $E(G) \subseteq \{ab: a \in A, \; b \in B \}$ .

For a digraph G and $v\in V(G)$ , we denote the set of outneighbours and inneighbours of v by $N_G^{+}(v)$ and $N_G^{-}(v)$ , respectively, and we write $d_G^{+}(v):=|N_G^{+}(v)|$ and $d_G^{-}(v):=|N_G^{-}(v)|$ for the out- and indegree of v, respectively. For $S\subseteq V(G)$ we write $d^{-}(v,S):= |N_G^-(v) \cap S|$ and $d^{+}(v,S):= |N_G^+(v) \cap S|$ . We write $\delta ^+(G)$ and $\delta ^-(G)$ , respectively, for the minimum out- and indegree of G and $\delta ^{0}(G):=\min \{\delta ^{+}(G),\delta ^{-}(G)\}$ for the minimum semi-degree. Similarly, the maximum semi-degree $\Delta ^{0}(G)$ of G is defined by $\Delta ^{0}(G):=\max \{\Delta ^{+}(G),\Delta ^{-}(G)\}$ where $\Delta ^{+}(G)$ and $\Delta ^{-}(G)$ denote the maximum out- and maximum indegree of G, respectively. A digraph is called d-regular if each vertex has exactly d outneighbours and d inneighbours.

The notation above extends to undirected graphs in the obvious ways. In particular for an undirected graph G, we write $\Delta (G)$ and $\delta (G)$ , respectively for the maximum degree and the minimum degree. A graph is called d-regular if each vertex has exactly d neighbours. For a vertex $v \in V(G)$ and subset $S \subseteq V(G)$ , we write $d_G(v,S):=|N_G(v) \cap S|$ .

A directed path Q in a digraph G is a subdigraph of G such that $V(Q) = \{v_1, \ldots , v_k \}$ for some $k \in \mathbb {N}$ and $E(Q) = \{v_1v_2, v_2v_3, \ldots , v_{k-1}v_k \}$ . We denote such a directed path by its vertices in order (i.e., we write $Q = v_1v_2 \cdots v_k$ ). A directed cycle in G is exactly the same except that it also includes the edge $v_kv_1$ . Sometimes we identify paths with their edge sets.

A set of vertex-disjoint directed paths $\mathcal {Q}=\{Q_1,Q_2,\ldots \}$ in a digraph G is called a path system in G. We interchangeably think of $\mathcal {Q}$ as a set of vertex-disjoint directed paths in G and as a subdigraph of G with vertex set $V(\mathcal {Q}) = \bigcup _{i} V(Q_i)$ and edge set $E(\mathcal {Q}) = \bigcup _{i} E(Q_i)$ . We sometimes call this subdigraph the graph induced by $\mathcal {Q}$ . A matching M in a digraph (or undirected graph) G is a set of edges $M \subseteq E(G)$ such that every vertex of G is incident to at most one edge in M. If $M = \{a_ib_i: i \in [m]\}$ is a matching in a digraph, then we write $V^+(M):= \{a_i : i \in [m]\}$ and $V^-(M) = \{ b_i : i \in [m] \}$ . A $1$ -factor in a digraph G is a set of vertex-disjoint directed cycles whose union has the same vertices as G.

Throughout the paper, we will work with partitions of a vertex set V of the form $\{V_{ij}: i,j \in [k]\}$ for some $k \in \mathbb {N}$ (so a partition into $k^2$ parts). For each i, we write $V_{i*} := V_{i1} \cup \cdots \cup V_{ik}$ and $V_{*i} := V_{1i} \cup \cdots \cup V_{ki}$ . Note that $\{V_{i*}: i \in [k]\}$ is a partition of V as is $\{V_{*i}: i \in [k]\}$ . Note also that $V_{ij} = V_{i*} \cap V_{*j}$ .

For two sets A and B, the symmetric difference of A and B is the set $A\triangle B := (A\setminus B) \cup (B\setminus A)$ . For $x, y \in (0,1]$ , we often use the notation $x \ll y$ to mean that x is sufficiently small as a function of y [i.e., $x \leq f(y)$ ] for some implicitly given non-decreasing function $f:(0,1] \rightarrow (0,1]$ . We implicitly assume all constants in such hierarchies are positive, and we omit floors and ceilings whenever this does not affect the argument.

We use the following non-standard notation. Let G be a directed graph and let $U,W \subseteq V(G)$ not necessarily disjoint. We define $G[U,W]$ to be the auxiliary undirected bipartite graph with bipartition $U , W$ where, for each $u \in U$ and $w \in W$ , $uv$ is an (undirected) edge of $G[U,W]$ if and only if $uv$ is a directed edge in $E(G)$ . Note that for each vertex in $U\cap W$ , there are two copies of the vertex in $G[U,W]$ which are viewed as distinct. So $G[U,W]$ has $|U| + |W|$ vertices and $e_G(U,W)$ edges.

3. Robust expanders

We first define robust expansion for graphs. Let $0 < \nu \le \tau < 1$ and let G be a graph on n vertices. For $S \subseteq V(G)$ , the $\nu $ -robust neighbourhood $RN_{\nu ,G}(S)$ is the set of all those vertices with at least $\nu n$ neighbours in S. We say that G is a robust $(\nu , \tau )$ -expander if every $S \subseteq V(G)$ with $\tau n \le |S| \le (1 - \tau ) n$ satisfies $|RN_{\nu ,G} (S)| \ge |S| + \nu n$ . In fact we will mainly be concerned with bipartite robust expanders. Let G be a bipartite graph with bipartition $A, B$ . We say that G is a bipartite robust $(\nu , \tau )$ -expander with bipartition $A, B$ if every $S \subseteq A$ with $\tau |A| \le |S| \le (1 - \tau ) |A|$ satisfies $|RN_{\nu ,G} (S) \cap B| \ge |S| + \nu n$ . Note that the order of A and B matters here. Our first lemma says that bipartite robust expansion is robust to small alterations; the lemma can easily be derived from the definition of robust expansion.

Next we give a structural result due to Kühn, Lo, Osthus and Staden [Reference Kühn, Lo, Osthus and Staden21] which states that any regular graph of linear minimum degree has a vertex partition into a small number of parts where each part induces a robust expander or a bipartite robust expander. In fact we state the special case of this result for bipartite graphs, which is all we require.

Theorem 3.2 (Bipartite special case of [Reference Kühn, Lo, Osthus and Staden21, Theorem 3.1])

For all $\alpha , \tau> 0$ and every non-decreasing function $f: (0, 1) \rightarrow (0, 1)$ , there exists $n_0$ such that the following holds. For all d-regular bipartite graphs G on $2n \geq n_0$ vertices with bipartition $A,B$ and $d \ge \alpha n$ , there exist $\rho , \nu $ with

$$ \begin{align*} 1/n_0 \le \rho \le \nu \le \tau; \qquad \rho \le f(\nu); \qquad \text{and} \qquad 1/n_0 \le f(\rho) \end{align*} $$

such that there is a partition of $V(G)$ into sets $A_1,\dots , A_k,B_1, \dots , B_k$ with the following properties for all $i \in [k]$ :

1. (i) $G[A_i \cup B_i]$ is a bipartite robust $(\nu , \tau )$ -expander with bipartition $A_i,B_i$ ;

2. (ii) for all $x \in A_i \cup B_i$ and $j \in [k]$ , $d(x, A_i \cup B_i) \ge d(x, A_j \cup B_j) $ , so in particular, $\delta ( G[ A_i \cup B_i ] ) \ge d/k$ ;

3. (iii) $\left | |A_i| - |B_i| \right | \le 2 \rho n$ ;

4. (iv) all but at most $2 \rho n $ vertices $x \in A_i \cup B_i$ satisfy $d(x, A_i \cup B_i) \ge d - 2 \rho n $ ;

5. (v) $k \le n / (d-2 \rho n) $ ;

6. (vi) $A_1, \ldots , A_k$ is a partition of A and $B_1, \ldots , B_k$ is a partition of B.

We now define robust expansion for digraphs. Let $0 < \nu \le \tau < 1$ and let G be a digraph on n vertices. For $S \subseteq V(G)$ , the $\nu $ -robust outneighbourhood $RN^+_{\nu ,G}(S)$ is the set of all those vertices with at least $\nu n$ inneighbours in S. We say that G is a robust $(\nu , \tau )$ -outexpander if every $S \subseteq V(G)$ with $\tau n \le |S| \le (1 - \tau ) n$ satisfies $|RN^+_{\nu ,G} (S)| \ge |S| + \nu n$ .

The proposition above follows immediately from the definitions; we crudely replace $(\nu , \tau )$ with $(\nu /2, 2\tau )$ to account for the loss of any edges of the form $a_ib_i$ .

We now state and prove a structure lemma for regular digraphs that is derived from Theorem 3.2. This will be one of the key ingredients in the proof of Theorem 1.3. It says that any dense regular digraph G has two vertex partitions $V(G)=V_{1*} \cup \cdots \cup V_{k*}$ and $V(G)=V_{*1} \cup \cdots \cup V_{*k}$ with k relatively small such that for each i, the (undirected) bipartite graph $G[V_{i*}, V_{*i}]$ is a bipartite robust expander (with bipartition $V_{i*}, V_{*i}$ ). Various other degree and size conditions relating to the partition are also given. Note that in the theorem below, we actually give a partition $\{V_{ij}: i,j \in [k]\}$ and recall that for each $i\in [k]$ , we write $V_{i*} := V_{i1} \cup \cdots \cup V_{ik}$ and $V_{*i} := V_{1i} \cup \cdots \cup V_{ki}$ . Thus $V_{ij} = V_{i*} \cap V_{*j}$ .

Theorem 3.5. For all $\alpha , \tau> 0$ and every non-decreasing function $f : (0, 1) \rightarrow (0, 1)$ , there exists $n_0$ such that the following holds. For all d-regular digraphs G on $n \ge n_0$ vertices with $d \ge \alpha n$ , there exist $\rho , \nu $ with

$$ \begin{align*} 1/n_0 \le \rho \le \nu \le \tau; \qquad \rho \le f(\nu); \qquad \text{and} \qquad 1/n_0 \le f(\rho) \end{align*} $$

such that there is a partition $\mathcal {P} = \{ V_{ij}: i,j \in [k]\}$ of $V(G)$ satisfying, for all $i,j \in [k]$ ,

1. (i) $G[V_{i*},V_{*i}]$ is a bipartite robust $(\nu , \tau )$ -expander with bipartition $V_{i*},V_{*i}$ ;

2. (ii) for all $x \in V_{ij}$ and $i',j' \in [k]$ , $d^+(x, V_{*i}) \ge d^+(x, V_{*i'})$ and $d^-(x, V_{j*}) \ge d^-(x, V_{j'*})$ , so in particular, $\delta ( G [ V_{i*},V_{*i} ] ) \ge d/k$ ;

3. (iii) $\left | |V_{i*}| - |V_{*i}| \right | \le 2\rho n$ ;

4. (iv) all but at most $2\rho n $ vertices $x \in V_{ij}$ satisfy $d^+(x, V_{*i}), d^-(x, V_{j*}) \ge d - 2\rho n $ ;

5. (v) $k \le n / (d-2\rho n)$ .

Proof. We simply apply Theorem 3.2 to the obvious bipartite graph obtained from the directed graph G, with the natural correspondence in parameters, as follows.

For any d-regular digraph G on n vertices, let $H_G$ be the d-regular bipartite undirected graph on $2n$ vertices defined as follows. Let $V(H_G) = A \cup B$ , where A and B are disjoint copies of $V(G)$ and for $a \in A$ and $b \in B$ let $ab \in E(H_G)$ if and only if $ab \in E(G)$ . Now applying Theorem 3.2 with $H_G$ playing the role of G, we obtain the following statement.

For all $\alpha , \tau> 0$ and every non-decreasing function $f: (0, 1) \rightarrow (0, 1)$ , there exists $n_0$ such that the following holds. For all d-regular digraphs G on $n \geq n_0$ vertices and $d \ge \alpha n$ , taking $H_G$ to be the corresponding bipartite d-regular graph on $2n \geq n_0$ vertices there exist $\rho , \nu $ with

$$ \begin{align*} 1/n_0 \le \rho \le \nu \le \tau; \qquad \rho \le f(\nu) ;\qquad \text{and} \qquad 1/n_0 \le f(\rho) \end{align*} $$

such that there is a partition of $V(H_G)$ into sets $A_1,\dots , A_{k}, B_1, \dots , B_{k}$ with the following properties for all $i \in [k]$ :

1. (i^′) $H_G[A_i \cup B_i]$ is a bipartite robust $(\nu , \tau )$ -expander with bipartition $A_i,B_i$ ;

2. (ii^′) for all $x \in A_i \cup B_i$ and $j \in [k]$ , $d_{H_G}(x, A_i \cup B_i) \ge d_{H_G}(x, A_j \cup B_j) $ , so in particular, $\delta ( H_G [ A_i \cup B_i ] ) \ge d/k$ ;

3. (iii^′) $\left | |A_i| - |B_i| \right | \le 2 \rho n$ ;

4. (iv^′) all but at most $2 \rho n $ vertices $x \in A_i \cup B_i$ satisfy $d_{H_G}(x, A_i \cup B_i) \ge d - 2 \rho n $ ;

5. (v^′) $k \le n / (d-2 \rho n)$ ;

6. (vi^′) $A_1, \ldots , A_k$ is a partition of A and $B_1, \ldots , B_k$ is a partition of B.

For $i,j \in [k]$ , define $V_{ij} = A_i \cap B_j$ (where we think of $A_i$ and $B_j$ as sets of vertices of the digraph G). First note that, by (vi $'$ ), $\mathcal {P} = \{V_{ij} : i,j \in [k] \}$ is a partition of $V(G)$ , and $A_i = V_{i*}$ and $B_i = V_{*i}$ . Now the natural correspondence between $H_G$ and G means that (i $'$ )–(v $'$ ) imply (i)–(v), respectively.

4. Finding long cycles

Theorem 3.5 from the previous section shows that every (dense) regular digraph has a vertex partition with some useful properties. In this section, we show how properties (i) and (ii) from Theorem 3.5 together with a simple balancing condition on the partition allow us to construct few long cycles that can cover all the vertices of several parts in the partition.

Let G be a digraph on V and let $\mathcal {P} =\{V_{ij}:i,j \in [k]\}$ be a partition of V where we allow some parts to be empty. We say that $\mathcal {P}$ is balanced if $|V_{i*}| = |V_{*i}|$ for all $i \in [k]$ . For $i, j \in [k]$ , let $G_{ij}$ be the subdigraph of G on $V_{i*} \cup V_{*j}$ with edges from $V_{i*}$ to $V_{*j}$ , that is, $E(G_{ij}) = E_G(V_{i*},V_{*j}) = E(G) \cap (V_{i*} \times V_{*j})$ .

Define $S(\mathcal {P})$ to be the graph (without loops) on $[k]$ such that for all $i,j\in [k]$ with $i\neq j$ , $ij \in E(S(\mathcal {P}))$ if and only if $V_{ij} \cup V_{ji} \ne \emptyset $ . The connected components of $S(\mathcal {P})$ will determine which parts of $\mathcal {P}$ can be covered by one long cycle. We remark that, by definition, $\bigcup _{i\in I}(V_{i*}\cup V_{*i})$ and $\bigcup _{j\in J}(V_{j*}\cup V_{*j})$ are disjoint for two distinct connected components I and J of $S(\mathcal {P})$ . The aim of this section is to prove the following lemma.

The lemma above will be used as follows. Suppose we have a dense d-regular digraph G with a vertex partition $\mathcal {P}$ such as that given by Theorem 3.5 but with the additional property that $\mathcal {P}$ is balanced. Then Lemma 4.1 applied to each connected component of $S(\mathcal {P})$ gives us a collection of s vertex-disjoint cycles that cover $V(G)$ , where s is the number of connected components in $S(\mathcal {P})$ . So in later sections we will be interested in obtaining balanced partitions $\mathcal {P}$ where the number of connected components of $S(\mathcal {P})$ is ‘small’.

We need the following theorem, which states that a robust outexpander with linear minimum degree contains a (directed) Hamilton cycle.

The next lemma shows that, under the conditions of Lemma 4.1, $V_{i*} \cup V_{*i}$ can be covered either by vertex-disjoint paths from $V_{i*} \setminus V_{ii}$ to $V_{*i} \setminus V_{ii}$ or by a cycle, and it is used inductively to prove Lemma 4.1.

We now prove Lemma 4.1. For a connected component I in $S(\mathcal {P})$ , the idea is to apply Lemma 4.3 to each $i \in I$ . Some care is needed to ensure that the union of path systems forms only one cycle.

Proof of Lemma 4.1

If I consists of a single vertex say i, then the result follows by Lemma 4.3 (since in that case $V_{i*} = V_{*i} = V_{ii}$ ). Now assume $|I|\geq 2$ , so $V_{i*}\cup V_{*i}\neq V_{ii}$ for all $i\in I$ . Without loss of generality, let $I = [\ell ]$ and order the indices of I such that for each $j \in [\ell -1]$ , there is a $j' \in [\ell ] \setminus [j]$ with $j j' \in E(S(\mathcal {P}))$ . This can be achieved since I is connected, (e.g., by taking a reverse breadth-first search ordering). Note that $V_{ij}=V_{ji}=\emptyset $ for all $i\in [\ell ]$ and $j\notin [\ell ]$ .

For $i\in [\ell ]$ , let $W^+_i=\bigcup _{i'\in [i],\, j'\notin [i]}V_{i'j'} = \bigcup _{i' \in [i]}V_{i'*} \setminus \bigcup _{i',j'\in [i]}V_{i'j'}$ and $W^-_i=\bigcup _{i'\notin [i],\, j'\in [i]}V_{i'j'} = \bigcup _{j' \in [i]}V_{*j'} \setminus \bigcup _{i',j'\in [i]}V_{i'j'}$ . Since $\mathcal {P}$ is balanced, $|W^+_i| = |W^-_i|$ for each $i \in [\ell ]$ . Also, by our ordering of the indices, we have that $W_i^+,W_i^-\neq \emptyset $ for $i\in [\ell -1]$ and that $W_{\ell }^+=W_{\ell }^-= \emptyset $ .

Let $\mathcal {Q}_0=W_0^+=W_0^-=\emptyset $ and suppose for some $i \in [\ell ]$ , we have already found a path system $\mathcal {Q}_{i-1}$ such that $V(\mathcal {Q}_{i-1}) = \bigcup _{i' \in [i-1]} \left ( V_{i'*} \cup V_{*i'} \right )$ and $\mathcal {Q}_{i-1}$ consists of precisely $|W^+_{i-1}|$ paths from $W^+_{i-1}$ to $W^-_{i-1}$ . We now construct $\mathcal {Q}_{i}$ as follows. Let $\phi : V_{i*} \setminus V_{ii} \rightarrow V_{*i} \setminus V_{ii} $ be any bijection such that if there is a path in $\mathcal {Q}_{i-1}$ from $v_+ \in V_{*i}\setminus V_{ii}$ to $v_- \in V_{i*}\setminus V_{ii}$ , then $\phi (v_-) = v_+ $ . Apply Lemma 4.3 and obtain a path system $\mathcal {Q}_i'$ in $G_{ii}$ such that $\mathcal {Q}_i' \cup \{\phi (v)v : v \in V_{i*}\setminus V_{ii}\}$ is a cycle with vertex set $V_{i*} \cup V_{*i}$ . We set $\mathcal {Q}_{i} = \mathcal {Q}_{i-1} \cup \mathcal {Q}_i'$ .

Suppose $i = \ell $ . Note that $W^-_{\ell -1} = V_{\ell *} \setminus V_{\ell \ell }$ and $W^+_{\ell -1} = V_{*\ell } \setminus V_{\ell \ell }$ . Hence $\mathcal {Q}_{\ell -1}$ consists of paths from $\phi (v) \in W_{\ell -1}^+$ to $v \in W_{\ell -1}^-$ , we have $V(\mathcal {Q}_{\ell -1}) = \bigcup _{i' \in [\ell -1]} \left ( V_{i'*} \cup V_{*i'} \right )= \bigcup _{i' \in [\ell ]} \left ( V_{i'*} \cup V_{*i'} \right ) \setminus V_{\ell \ell }$ . Since $\mathcal {Q}_{\ell }' \cup \{\phi (v)v : v \in V_{\ell *} \setminus V_{\ell \ell }\}$ is a cycle with vertex set $V_{\ell *} \cup V_{* \ell }$ , we deduce that $\mathcal {Q}_{\ell } = \mathcal {Q}_{\ell }' \cup \mathcal {Q}_{\ell -1}$ is a cycle with vertex set $\bigcup _{i' \in [\ell ]} \left ( V_{i'*} \cup V_{*i'} \right )$ as required.

Suppose $i \in [\ell - 1]$ . Note that $V(\mathcal {Q}_{i}) = \bigcup _{i' \in [i]} \left ( V_{i'*} \cup V_{*i'} \right )$ . It remains to check that $\mathcal {Q}_i$ is a path system consisting of precisely $|W_i^+|$ paths from $W_i^+$ to $W_i^-$ . Note first that paths in $\mathcal {Q}^{\prime }_i \subseteq G_{ii}$ start in $V_{i*} \setminus V_{ii}$ and end in $V_{*i} \setminus V_{ii}$ and have all internal vertices in $V_{ii}$ . On the other hand a path in $\mathcal {Q}_{i-1}$ can only intersect $V_{*i}$ at its start point and $V_{i*}$ at its end point and it avoids $V_{ii}$ . Therefore in $\mathcal {Q}_{i} = \mathcal {Q}_{i-1} \cup \mathcal {Q}^{\prime }_i$ , all indegrees and outdegrees are at most $1$ . A vertex has indegree zero in $\mathcal {Q}_{i-1}\cup \mathcal {Q}_i'$ if and only if it is a start point of some path in $\mathcal {Q}_{i-1}\cup \mathcal {Q}_i'$ but not an endpoint of any path in $\mathcal {Q}_{i-1}\cup \mathcal {Q}_i'$ , (i.e., the set of vertices of indegree zero is

$$ \begin{align*} \left(W_{i-1}^+\cup (V_{i*}\setminus V_{ii})\right)\setminus \left(W_{i-1}^-\cup (V_{*i}\setminus V_{ii})\right)=\left(W_{i-1}^+\cup V_{i*}\right)\setminus V_{*i} =W_i^+ \end{align*} $$

and similarly the set of vertices of outdegree zero is $W_i^-$ . Finally it remains to check that there are no cycles in $\mathcal {Q}_{i-1}\cup \mathcal {Q}_i'$ . By our choice of $\phi $ , if there is a cycle, it spans all the vertices of $V(\mathcal {Q}_{i-1}\cup \mathcal {Q}_i')$ , but this cannot happen because vertices in $W_i^+\neq \emptyset $ have in-degree zero.

5. Balanced path systems and path contraction

As mentioned in the previous section, in order to apply Lemma 4.1 to obtain a suitable cycle partition of a dense regular digraph G, we will require a vertex partition $\mathcal {P}$ of G such as that given by Theorem 3.5 but with the additional properties that $\mathcal {P}$ is balanced and $S(\mathcal {P})$ has few connected components. In this section, we state a result (Lemma 5.4) that allows us to adjust a partition $\mathcal {P}$ to have these additional properties. In particular, if $\mathcal {P}$ is not balanced, Lemma 5.4 guarantees us a so-called $\mathcal {P}$ -balanced path system in G whose ‘contraction’ makes $\mathcal {P}$ balanced in a suitable way. At the end of the section we show how Lemma 5.4 implies our main result, Theorem 1.3.

Let $\mathcal {P} =\{V_{ij}:i,j \in [k]\}$ be a partition of a vertex set V and let G be a digraph on V. Recall the definition of $G_{ij}$ and of P being balanced at the beginning of Section 4. Write $G_{i*} = \bigcup _{j\neq i} G_{ij}$ and $G_{*j} = \bigcup _{i \ne j} G_{ij}$ . Let $\mathcal {B}(G, \mathcal {P}) = G - \bigcup _{i \in [k]} G_{ii}$ . Note that $\mathcal {B}(G, \mathcal {P}) = \bigcup _{i \in [k]} G_{i*} = \bigcup _{j \in [k]} G_{*j}$ . We say that a digraph H on V (where H is usually a path system in G) is $\mathcal {P}$ -balanced if, for all $i \in [k]$ ,

$$ \begin{align*} |V_{i*}|-|V_{*i}| = e_H(V_{i*},V) - e_H(V,V_{*i}) = \sum_{j \in [k]} e(H_{ij}) - \sum_{j \in [k]} e(H_{ji}) = e(H_{i*}) - e(H_{*i}). \end{align*} $$

The above gives three equivalent conditions for H to be $\mathcal {P}$ -balanced. Note that if H is regular, then it is $\mathcal {P}$ -balanced for any $\mathcal {P}$ .

Let Q be a path in G from $v_+ \in V_{i_+j_+}$ to $v_- \in V_{i_-j_-}$ . We define the Q-contracted subgraph $G'$ of G to be $G \setminus V(Q)$ together with a new vertex w such that $N^+_{G'}(w) = N^+_G(v_-) \setminus V(Q)$ and $N^-_{G'}(w) = N^-_G(v_+) \setminus V(Q)$ . We call $\mathcal {P}'=\{V_{ij}':i,j \in [k]\}$ the Q-contracted partition of $\mathcal {P}$ , where

(5.1) $$ \begin{align} V_{ij}' = \begin{cases} ( V_{ij} \setminus V(Q) ) \cup \{w\} & \text{if }(i,j) = (i_-,j_+);\\ V_{ij} \setminus V(Q) &\text{otherwise}. \end{cases} \end{align} $$

Let $\mathcal {Q}$ be a path system in G. The $\mathcal {Q}$ -contracted subgraph of G (and $\mathcal {Q}$ -contracted partition of $\mathcal {P}$ ) is obtained by successively contracting each $Q \in \mathcal {Q}$ for G (and $\mathcal {P}$ , respectively).

The following two propositions follow from the definition of $\mathcal {Q}$ -contraction. In fact the definitions are chosen precisely so that these propositions hold.

Proof. Let $Q \in \mathcal {Q}$ be a path from $v_+ \in V_{i_+j_+}$ to $v_- \in V_{i_-j_-}$ . For each $i,j \in [k]$ , let $q_{ij} = e(Q_{ij})$ (i.e., the number of edges of Q from $V_{i*}$ to $V_{*j}$ .) Let $\mathcal {P}' = \{ V_{ij}' : i,j \in [k] \} $ be the Q-contracted partition of $\mathcal {P}$ . Consider $i \in [k]$ . Note that

Therefore, $|V^{\prime }_{i*}| - |V^{\prime }_{*i}| = |V_{i*}| - |V_{*i}| - \sum _{j \in [k]} (q_{ij} - q_{ji}) = |V_{i*}| - |V_{*i}| - ( e(Q_{i*}) - e(Q_{*i}) )$ . A similar statement holds for the $\mathcal {Q}$ -contracted partition $\mathcal {P}^* = \{ V_{ij}^* : i,j \in [k] \}$ of $\mathcal {P}$ , and the result follows as $\mathcal {Q}$ is $\mathcal {P}$ -balanced (i.e., for each $i \in [k]$ , $ |V^*_{i*}| - |V^*_{*i}|=|V_{i*}| - |V_{*i}| - ( e(\mathcal {Q}_{i*}) - e(\mathcal {Q}_{*i}) )=0$ ).

The next lemma says that one can adjust the vertex partition $\mathcal {P}$ found in Theorem 3.5 by a small amount, and find a $\mathcal {P}$ -balanced path system $\mathcal {Q}$ such that $S(\mathcal {P}^*)$ has few connected components, where $\mathcal {P}^*$ is the $\mathcal {Q}$ -contracted partition of $\mathcal {P}$ . It is exactly what we need to prove Theorem 1.3, as we shall see. In Section 6, we break this lemma down into two further lemmas.

5.1. Proof of Theorem 1.3

We now prove Theorem 1.3 assuming Lemma 5.4.

Proof of Theorem 1.3

Choose a non-decreasing function $g : (0, 1) \rightarrow (0, 1)$ such that the requirements of Lemmas 4.1 (with $m = \alpha n/2$ ) and 5.4 are satisfied whenever $n, \gamma , \nu , \tau , \alpha $ satisfy

$$ \begin{align*} 1/n \ll_g \gamma \ll_g \nu \leq \tau \ll_g \alpha, \end{align*} $$

where we write $a \ll _g b$ to mean $a \leq g(b)$ . Set $f(x) := \min (x, g(x^2/4))$ . Fix $\tau \leq f(\alpha )$ and apply Theorem 3.5 to obtain $n_0$ such that for any d-regular digraph G on $n \geq n_0$ vertices with $d \ge \alpha n $ , there exists $\gamma $ and $\nu $ such that

$$ \begin{align*} 1/n \ll_f \gamma \ll_f \nu \leq \tau \ll_f \alpha \end{align*} $$

and a partition $\mathcal {P} = \{ V_{ij}: i,j \in [k]\}$ of $V(G)$ satisfying, for all $i,j \in [k]$ ,

1. (a₁) $G[V_{i*},V_{*i}]$ is a bipartite robust $(\nu , \tau )$ -expander with partition $V_{i*},V_{*i}$ ;

2. (a₂) for all $x \in V_{ij}$ and $i',j' \in [k]$ , $d^+(x, V_{*i}) \ge d^+(x, V_{*i'})$ and $d^-(x, V_{j*}) \ge d^-(x, V_{j'*})$ , so in particular, $\delta ( G [ V_{i*},V_{*i} ] ) \ge d/k$ ;

3. (a₃) $\left | |V_{i*}| - |V_{*i}| \right | \le \gamma n$ ;

4. (a₄) all but at most $\gamma n $ vertices $x \in V_{ij}$ satisfy $d^+(x, V_{*i}), d^-(x, V_{j*}) \ge d - \gamma n $ ;

5. (a₅) $k \le n / (d-\gamma n)$ .

Note that ( $\mathrm{a}_5$ ) in particular implies that $\gamma \ll 1/k$ . Furthermore, ( $\mathrm{a}_2$ ) and ( $\mathrm{a}_4$ ) together with $\gamma \ll 1/k$ imply that

1. (a₆) for all $i \in [k]$ , $|V_{i*}|,|V_{*i}|\ge d-\gamma n$ .

Let $q=2$ if G is oriented, and $q=1$ otherwise. Apply Lemma 5.4 and obtain a partition $\mathcal {P}' = \{ V^{\prime }_{ij} : i,j \in [k]\}$ of $V(G)$ and a $\mathcal {P}'$ -balanced path system $\mathcal {Q}$ in G such that

1. (b₁) for all $i, j \in [k]$ , $| V_{ij} \triangle V^{\prime }_{ij}| \le \gamma ^{1/2} n$ ;

2. (b₂) for all $i \in [k], \delta (G[V_{i*}', V_{*i}']) \geq (d/k) - \gamma ^{1/2}n$ ;

3. (b₃) $ e(\mathcal {Q}) \le \gamma ^{1/2} n$ ;

4. (b₄) $S(\mathcal {P}^*)$ has at most $n/(qd+1)$ connected components, where $\mathcal {P}^*$ is the $\mathcal {Q}$ -contracted partition of $\mathcal {P}'$ .

Let $\mathcal {P}^* = \{V^*_{ij}:i,j \in [k]\}$ and let $G^*$ be the $\mathcal {Q}$ -contracted subgraph of G. We will check that, for all $i, j \in [k]$ ,

1. (c₁) $\mathcal {P}^*$ is balanced;

2. (c₂) $| V_{ij} \triangle V^*_{ij}| \le 5 \gamma ^{1/2} n$ ;

3. (c₃) $|V_{i*}^*|,|V_{*i}^*|\ge d- 6 k\gamma ^{1/2} n \ge d/2$ ;

4. (c₄) $G^*[V^*_{i*},V^*_{*i}]$ is a bipartite robust $(\nu /2, 2\tau )$ -expander with bipartition $V^*_{i*},V^*_{*i}$ ;

5. (c₅) $\delta ( G^*[V^*_{i*},V^*_{*i}] ) \ge \alpha ^2 |V^*_{i*}|/2$ .

Note that ( $\mathrm{c}_1$ ) holds by Proposition 5.3. Consider $i,j \in [k]$ . Note that

implying ( $\mathrm{c}_2$ ). Hence ( $\mathrm{c}_3$ ) follows from ( $\mathrm{a}_6$ ) and ( $\mathrm{c}_2$ ) as $\gamma \ll \alpha ,1/k$ . By ( $\mathrm{a}_1$ ), ( $\mathrm{c}_2$ ), ( $\mathrm{c}_3$ ), and $\gamma \ll \nu $ , Lemma 3.1 (applied to the bipartite graph $G^*[V^*_{i*},V^*_{*i}]\cup G[V_{i*},V_{*i}]$ with $V_{i*}, V_{*i}, V^*_{i*}, V^*_{*i}$ playing the roles of $A,B, A', B'$ respectively) implies that ( $\mathrm{c}_4$ ) holds. By ( $a_6$ ), ( $\mathrm{b}_1$ ), and ( $\mathrm{b}_3$ ), we have $e(\mathcal {Q})\leq \gamma ^{1/2}n<d-\gamma n-k\gamma ^{1/2}n\leq |V^{\prime }_{i*}|,|V^{\prime }_{*i}|$ as $\gamma \ll \alpha ,1/k$ . Hence, by Proposition 5.2,

where the last inequality holds as $\gamma \ll \alpha , 1/k$ and $k (\alpha - \gamma )\leq 1$ by ( $\mathrm{a}_5$ ). Hence ( $\mathrm{c}_5$ ) holds.

Apply Lemma 4.1 (with $G^*$ , $\mathcal {P}^*$ , $d/2$ , $\nu /2$ , $2\tau $ , $\alpha ^2/2$ playing the roles of G, $\mathcal {P}$ , m, $\nu $ , $\tau $ , $\alpha $ , respectively) to each connected component I in $S(\mathcal {P}^*)$ . Thus, by ( $\mathrm{b}_4$ ), $G^*$ can be covered with at most $n/(qd + 1)$ vertex-disjoint cycles (where we have a cycle on $\cup _{i \in I} (V_{i*} \cup V_{*i})$ for each component I of $S(\mathcal {P}^*)$ so that each cycle has length at least $d/2$ by ( $\mathrm{c}_3$ )). By Proposition 5.1, G can be covered with at most $n/(qd+1)$ vertex-disjoint cycles (each of length at least $d/2$ ).

6. Proof of Lemma 5.4

In the last section we reduced the task of proving Theorem 1.3 to the task of proving Lemma 5.4. In this section we reduce this further to the task of proving Lemmas 6.1 and 6.3. We start by introducing the notion of a non-trivial path system, which will help us control the number of connected components of $S(\mathcal {P}^*)$ (as required in the conclusion of Lemma 5.4).

Recall the definitions at the beginning of Sections 4 and 5. Let $\mathcal {P} =\{V_{ij}:i,j \in [k]\}$ be a partition of a vertex set V. We call a path system $\mathcal {Q}$ on V non-trivial if there exists $i \ne j$ such that $V_{ij} \setminus V(\mathcal {Q}) \ne \emptyset $ or $\mathcal {Q}$ contains a path from $V_{*j}$ to $V_{i*}$ . Otherwise, we call $\mathcal {Q}$ a trivial path system. Note that if $\mathcal {Q}$ is non-trivial, then the $\mathcal {Q}$ -contracted partition $\mathcal {P}'$ of $\mathcal {P}$ has the property that $S(\mathcal {P}')$ has at least one edge so that its number of connected components is strictly less than its number of vertices.

Lemma 6.1 below shows that, under the same hypothesis as Lemma 5.4, there exists a $\mathcal {P}$ -balanced path system with few edges, and moreover that this path system is non-trivial if further assumptions are made. We shall see at the end of the section that the existence of such a non-trivial $\mathcal {P}$ -balanced path system is enough to prove Lemma 5.4. In fact, a $\mathcal {P}$ -balanced path system alone (without the condition of being non-trivial) is enough to prove Lemma 5.4 except in the extremal cases when $d \approx n/k$ if G is digraph and when $d \approx n/ 2k$ if G is an oriented graph.

In case Lemma 5.4 gives a $\mathcal {P}$ -balanced but trivial path system (i.e., when both (m1) and (m2) fail), we use the next proposition to slightly modify $\mathcal {P}$ and Lemma 6.3 to find the desired non-trivial, $\mathcal {P}$ -balanced path system.

Proof. For each $i \in [k]$ , let $V_{ii}' =V_{i*}$ . Clearly, $\mathcal {P}' = \{V_{ii}':i \in [k]\}$ is a partition of $V(G^0)$ . Note that (ii) implies (i $'$ ). For $v \in V_{ij}$ (possibly $i=j$ ), note that

If $i\neq j$ , similarly, we have

Also, for all $v\in V_{ii}$ , (i) implies $d^{-}(v, V_{ii}')= d^{-}(v, V_{i*})\ge d/k$ . Hence (ii $'$ ) holds.

Before we can prove Lemma 5.4, we need a good lower bound in oriented graphs for $|\bigcup _{i \in I} \left ( V_{i*} \cup V_{*i} \right )|$ for any connected component I in $S(\mathcal {P})$ ; see Proposition 6.5. For this, we use the following result on the minimum semi-degree threshold for an oriented graph to be a robust outexpander. Recall that the definition of robust outexpander is given immediately after Remark 3.3.

Proposition 6.5. Let $1/ n \ll \gamma \ll \alpha , 1/k$ . Let G be an oriented graph on n vertices with $\Delta ^0(G)\le d$ where $ d \ge \alpha n$ . Let $\mathcal {P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G)$ such that $|V_{i*}| \ge d/2$ for all $i \in [k]$ . Suppose that, for all $i,j \in [k]$ , all but at most $\gamma n$ vertices $x\in V_{ij}$ satisfy $d^+(x, V_{*i}), d^-(x, V_{j*}) \ge d - \gamma n $ . Let $I \subseteq S(\mathcal {P})$ be a connected component. Then

$$ \begin{align*} \left| \bigcup_{i \in I } (V_{i*} \cup V_{*i}) \right| \ge \begin{cases} 2(d - \gamma n) & \text{if }|I| =1,\\ 9(d - 5\gamma n)/4 & \text{otherwise.} \end{cases} \end{align*} $$

Proof. Without loss of generality, let $I = [\ell ]$ . For each $i \in [\ell ]$ , we can find vertices $ v_i \in V_{i*}$ such that $d^+(v_i, V_{*i})\ge d - \gamma n $ as $\gamma \ll \alpha ,1/k$ , which shows in particular that $|V_{*i}|\geq d-\gamma n$ . Hence, if $\ell \ge 3$ ,

$$ \begin{align*} \left|\bigcup_{i \in I } (V_{i*} \cup V_{*i})\right| \ge \sum_{i \in [\ell] } | V_{*i}| \ge \ell (d - \gamma n) \ge 3(d-\gamma n)\ge 9(d-5\gamma n)/4. \end{align*} $$

If $\ell = 1$ , then $V_{11} = V_{1*} = V_{*1}$ . There exists a vertex $v \in V_{11}$ such that $d^+(v, V_{11}), d^-(v, V_{11}) \ge d - \gamma n $ . Since G is an oriented graph, we have $|V_{11}| \ge d^+(v, V_{11}) + d^-(v, V_{11}) \ge 2(d - \gamma n)$ .

If $\ell = 2$ , then $\bigcup _{i \in [2] } V_{i*} = \bigcup _{i \in [2] } V_{*i}$ and $V_{ij} = V_{ji} = \emptyset $ for all $(i,j) \in [2] \times ([k] \setminus [2])$ . Let $V_I = \bigcup _{i \in [2] } V_{i*}=\bigcup _{i \in [2] } V_{*i}$ . There exists a vertex subset $W \subseteq V_I$ such that

$$ \begin{align*} |W| \ge |V_I| - 4 \gamma n \ge 2(d-\gamma n)-4\gamma n \ge \alpha n, \end{align*} $$

and such that, writing $W_{ij} = W \cap V_{ij}$ for $i,j \in [2]$ , we have for all $i,j \in [2]$ that

(6.1) $$ \begin{align} d^+(x, W_{*i}), d^-(y, W_{j*}) \ge d - 5\gamma n \text{ for all } x \in W_{i*} \text{ and } y \in W_{*j}. \end{align} $$

Hence $\delta ^0( G[W] ) \ge d - 5\gamma n$ .

Let $\tau $ be a constant with $\gamma \ll \tau \ll \alpha $ . Next we show that $G[W]$ is not a robust $( \gamma ^{1/3} , \tau )$ -outexpander. For $i \in [2]$ , (6.1) implies that $|W_{i*}| \geq d - 5 \gamma n \ge \tau n \ge \tau |W|$ and so $\tau |W| \leq |W_{i*}| \leq (1 - \tau )|W|$ . Since $W_{1*} \cup W_{2*} = W_{*1} \cup W_{*2}$ , we may assume without loss of generality that $|W_{1*}| \geq |W_{*1}|$ . Recall that $\Delta ^0(G) \le d$ . By (6.1), each vertex in $|W_{1*}|$ has at most $5\gamma n$ outneighbours in $W_{*2}$ , so

$$ \begin{align*} |\mathrm{RN}_{\gamma^{1/3}}^+(W_{1*}) \cap W_{*2}| \leq \frac{e_G(W_{1*}, W_{*2})}{ \gamma^{1/3} |W|} \le \frac{5 \gamma n^2} {\alpha \gamma^{1/3} n } < \gamma^{1/3} \alpha n \le \gamma^{1/3} |W|, \end{align*} $$

where the penultimate inequality holds as $\gamma \ll \alpha $ . This implies that

$$ \begin{align*} |\mathrm{RN}_{\gamma^{1/3}}^+(W_{1*})| \le |W_{*1}| + |\mathrm{RN}_{\gamma^{1/3}}^+(W_{1*}) \cap W_{*2}| < |W_{1*}| + \gamma^{1/3} |W|. \end{align*} $$

Hence $G[W]$ is not a robust $( \gamma ^{1/3}, \tau )$ -outexpander as claimed. Lemma 6.4 with $\varepsilon =5/72$ implies $\delta ^0( G[W] ) \leq (3/8 + \varepsilon )|W| = 4|W|/9$ . Then $4|W|/9 \geq d - 5\gamma n$ , and the result follows.

We now prove Lemma 5.4 assuming Lemmas 6.1 and 6.3.

Proof of Lemma 5.4

Let G and $\mathcal {P} = \{V_{ij}: i,j \in [k]\}$ be as in the statement of Lemma 5.4. We apply Lemma 6.1 (with $G^0 = G$ ) and obtain a $\mathcal {P}$ -balanced path system $\mathcal {Q}^0$ such that $e(\mathcal {Q}^0) \le k^2 \gamma n $ . Let $\mathcal {P}^{0}$ be the $\mathcal {Q}^0$ -contracted partition of $\mathcal {P}$ . Let $k^*$ be the smallest integer larger than $n/(qd+1)$ , that is, $k^* = \lfloor n/(qd+1) \rfloor +1 $ . If $S(\mathcal {P}^{0})$ has at most $n/(qd+1)$ connected components, then we are done by setting $\mathcal {P}' = \mathcal {P}$ and $\mathcal {Q}= \mathcal {Q}^0$ since $e(\mathcal {Q}^0) \le k^2 \gamma n\le \gamma ^{1/2} n $ . Hence we may assume that $S(\mathcal {P}^{0})$ has at least $k^*$ connected components.

Claim 6.6. We have $k^* = k$ and $S(\mathcal {P}^{0})$ is an empty graph on $[k^*]$ .

Proof of claim

Since $1/n\ll \gamma \ll \alpha $ and $d \ge \alpha n$ , we have that

(6.2) $$ \begin{align} \dfrac{n}{q(d-\gamma n)}-\dfrac{n}{qd+1}\leq \dfrac{1}{10}\text{ and so } \left\lfloor \frac{n}{ q (d-\gamma n ) } \right\rfloor \le \left\lfloor \frac{n}{q d+1} \right\rfloor +1 =k^*. \end{align} $$

First suppose that G is not oriented, so $q=1$ . Recall (iii) that $|V_{i*}| \ge d - \gamma n $ for all $i \in [k]$ , so $n\geq k(d-\gamma n)$ . Together with (6.2), we have $k \le \left \lfloor n/ (d-\gamma n) \right \rfloor \le k^*$ . On the other hand, since $S(\mathcal {P}^{0})$ has at least $k^*$ connected components, we have $k = | V( S(\mathcal {P}^{0}) ) | \ge k^*$ . Thus $k = k^*$ . Furthermore, $S(\mathcal {P}^{0})$ is an empty graph on $[k^*]$ (or else $S(\mathcal {P}^{0})$ would have fewer than $k^*$ connected components).

Next, suppose that G is oriented, so $q=2$ . Let $I_1 , \dots , I_{a+b}$ be connected components of $S(\mathcal {P}^{0})$ such that $|I_j| = 1$ if and only if $j \in [a]$ . Note that $ a+b \ge k^*$ and $\{ \bigcup _{i \in I_j } (V_{i*} \cup V_{*i}) : j \in [a+b] \}$ is a partition of $V(G)$ . Recall (iv) that for all $i,j\in [k]$ , all but at most $\gamma n $ vertices $x \in V_{ij}$ satisfy $d^+(x, V_{*i}), d^-(x, V_{j*}) \ge d - \gamma n $ . Also, we have that $|V_{i*}|\geq d-\gamma n\geq d/2$ for all $i\in [k]$ as $\gamma \ll \alpha $ . By Proposition 6.5, we have

$$ \begin{align*} n = \sum_{j \in [a+b]} \left| \bigcup_{i \in I_j } (V_{i*} \cup V_{*i}) \right| \ge 2(d-\gamma n) a + \frac{9(d-5\gamma n)}4 b =2(d-\gamma n)(a+b)+\dfrac{d-37\gamma n}{4}b. \end{align*} $$

Hence (6.2) implies that

$$ \begin{align*} \dfrac{1}{10}\geq \dfrac{n}{2(d-\gamma n)}-\dfrac{n}{2d+1}>\dfrac{n}{2(d-\gamma n)}-k^*\geq (a+b-k^*)+\dfrac{b(d-37\gamma n)}{8(d-\gamma n)}>(a+b-k^*)+\dfrac{b}{9}, \end{align*} $$

where the last inequality holds as $\gamma \ll \alpha $ . Recalling that $a+b \geq k^*$ , this shows that $a+b=k$ and $b = 0 $ , and so $ a = k^*$ , proving the claim.

Since $S(\mathcal {P}^{0})$ is an empty graph on $[k^*]$ , we deduce that $\mathcal {Q}^0$ is trivial as defined at the start of the section. By the moreover statement of Lemma 6.1, we have

1. $(\overline{\mathrm{m}1})\ \ \!\sum _{i,j \in [k] \colon i \ne j} |V_{ij}| \le k^2\gamma n \le \gamma ^{1/4} n$ and

2. $(\overline{\mathrm{m}2})$ for all distinct $i,j \in [k]$ and $ v \in V_{ij}$ , we have $ d^+(v,V_{*i}) - d^-(v, V_{i *}) \le 100k^{12} \gamma ^{1/3}d \le \gamma ^{1/4} n$ .

By Proposition 6.2 (with $\gamma ^{1/4}$ playing the role of $\gamma $ ), there exists a partition $\mathcal {P}' = \{V_{ii}':i \in [k]\}$ of $V(G)$ (and we take $V_{ij}' = \emptyset $ for $i \not = j$ ) such that, for all $i \in [k]$ ,

1. (i^′) $|V_{ii} \triangle V^{\prime }_{ii}| \le \gamma ^{1/4} n $ ;

2. (ii^′) $\delta ^0( G [ V_{ii}' ] ) \ge (d/k) - \gamma ^{1/4} n $ .

Finally, apply Lemma 6.3 (with $\mathcal {P}'$ playing the role of $\mathcal {P}$ ) by noting that property (i) of Lemma 6.3 holds by property (iii) of Lemma 5.4 together with ( $\overline {\text {m}1}$ ) and (i $'$ ) above, and properties (ii) and (iii) of Lemma 6.3 hold by Claim 6.6 and (ii $'$ ) above, respectively. Thus, in G, we obtain a non-trivial $\mathcal {P}'$ -balanced path system $\mathcal {Q}$ with $ e (\mathcal {Q}) \le k \leq \gamma ^{1/2}n$ . Also, if $\mathcal {P}^*$ is the $\mathcal {Q}$ -contracted partition of $\mathcal {P}'$ , then by Claim 6.6, $S(\mathcal {P}^*)$ has at most $k-1=k^* - 1 \leq n/(qd + 1)$ connected components since $\mathcal {Q}$ is non-trivial. This gives (c) and (d) in Lemma 5.4, while ( $\overline {\text {m}1}$ ) and (i $'$ ) imply (a), and (ii $'$ ) implies (b).

7. Proof of Lemma 6.1

Recall that in Lemma 6.1, we are given a d-regular digraph $G^0$ on n vertices together with a partition $\mathcal {P}=\{V_{ij}:i,j\in [k]\}$ of $V(G^0)$ satisfying certain size and degree conditions, and we must find a $\mathcal {P}$ -balanced path system $\mathcal {Q}$ with few edges. Moreover, we require that $\mathcal {Q}$ is non-trivial under certain circumstances. We begin with a sketch of our proof approach for this lemma (ignoring the moreover part for now). Recall the definitions at the start of Sections 4 and 5.

Let $V^+$ and $V^-$ be two disjoint copies of $V(G^0)$ . Let $V_{1*}^+, \dots , V_{k*}^+$ to be the partition of $V^+$ corresponding to $V_{1*}, \dots , V_{k*}$ in $V(G^0)$ . Similarly, let $V_{*1}^-, \dots , V_{*k}^-$ to be the partition of $V^-$ corresponding to $V_{*1}, \dots , V_{*k}$ in $V(G^0)$ . Let H be the bipartite digraph with bipartition $V^+, V^-$ such that for each $x \in V^+_{i*}$ and $y \in V^-_{*j}$ , we have $xy \in E(H)$ if and only if $xy \in E(G^0)$ and $i \ne j$ . So there is a bijection between $E(H)$ and $E(\mathcal {B}( G^0, \mathcal {P}))$ .

A naive approach is to find, for each $i, j \in [k]$ with $i \ne j$ , a matching $M_{ij}$ in $H[V_{i*}^+, V^-_{*j}]$ with $e(M_{ij}) = e(G^0_{ij})/d$ . If $Q_{ij}$ is the subgraph in $G^0$ corresponding to $M_{ij}$ , then $Q = \bigcup _{i,j \in [k]} Q_{ij}$ is $\mathcal {P}$ -balanced by construction since $G^0$ is d-regular. It is relatively easy to guarantee Q has few edges by first passing from $G^0$ to a suitable subdigraph G (see Proposition 7.6). However, there are three problems with this approach: (i) $\Delta ^0(Q)$ might be greater than one (meaning that Q is not a path system), (ii) $e(G^0_{ij})/d$ may not be an integer, and (iii) Q may contain cycles, even if $\Delta ^0(Q) \leq 1$ .

To overcome (i) and (ii), we consider a suitable flow problem by converting H to a network ${\mathcal {F}^* = (F^*,w, s^*, t^*)}$ as follows (the formal definition of a network and flow are stated in Section 7.1). Starting with the graph H (with all edges of H having capacity 1), we add new vertices $s^*, s_1, \dots , s_k, t^*, t_1, \dots , t_k$ , where $s^*$ and $t^*$ are the source and sink respectively and where the $s_i$ and $t_j$ are viewed as ‘local’ sources and sinks, respectively. For each $i,j \in [k]$ and each $x^+ \in V^+_{i*}$ , $y^- \in V^-_{*j}$ , we add the edges $s_i x^+$ and $y^- t_j$ , each of capacity $1$ . For each $i \in [k]$ , we add the edge $s^* s_i$ of capacity $\max \{|V_{i*}| - |V_{*i}|, 0\}$ , the edge $t_i t^*$ of capacity $\max \{|V_{*i}| - |V_{i*}|, 0\}$ and the edge $t_i s_i$ of infinite capacity. This gives the network $\mathcal {F}^*$ ; see Figure 1. Consider a maximum integer flow f for $\mathcal {F}^*$ . Define Q to be the subdigraph of $\mathcal {B}(G^0, \mathcal {P})$ such that $xy \in E(Q)$ if and only if $x^+ y^-$ has a flow of one in f, where $x^+$ and $y^-$ are the corresponding copies of x and y in $V^+$ and $V^-$ , respectively. Note that $\Delta ^0(Q) \le 1$ by construction. Also it is easy to check that Q is $\mathcal {P}$ -balanced provided all edges at $s^*$ (and hence also at $t^*$ ) are saturated by f.

Figure 1 The network $\mathcal {F}^*$ .

To guarantee this latter condition on f, it turns out that it is enough to find a fractional flow $f^*$ such that $f_{ij}^*$ , the total amount of flow in $f^*$ through the edges in $H[V_{i*},V_{*j}]$ , is roughly $e(G^0_{ij})/d$ for all $i\neq j$ . Such a fractional flow is almost a maximum flow, and we can use the max-flow min-cut theorem to convert this fractional flow into an integer flow that saturates the edges at $s^*$ (and $t^*$ ); see Claim 7.10. This also addresses (ii) above.

To deal with (iii), suppose we could find a matching $M^0$ of $G^0$ such that $e(M^0[V_{i*},V_{*j}]) \ge e(G^0_{ij})/d$ for all $i,j \in [k]$ . Then defining H (and $\mathcal {F}^*$ ) using $M^0$ instead of $G^0$ , we could apply the same argument as before to obtain a $\mathcal {P}$ -balanced subgraph $Q \subseteq M^0$ (which is therefore necessarily a path system as required). However, it is not always possible to find such a matching, so instead, for each $i,j$ , we find a large ‘extendable matching’ in $G^0_{ij}$ which consists of a matching $M_{ij} \subseteq G^0_{ij}$ together with sets of vertices $X_{ij}^+ \subseteq V_{i*}$ and $X_{ij}^- \subseteq V_{*j}$ where each vertex $v^+ \in X_{ij}^+$ has high outdegree in $G^0_{ij}$ and each vertex $v^- \in X_{ij}^-$ has high indegree in $G^0_{ij}$ , and where $\bigcup _{i,j} M_{ij}$ is a matching (see Lemma 7.8). Here the very large degrees of vertices in $X_{ij} = X_{ij}^+ \cup X_{ij}^-$ allow us to greedily extend $M_{ij}$ into a suitably large path systemFootnote ⁵ (see Proposition 7.9). These path systems would be disjoint (as required) if the $X_{ij}$ are disjoint, but a priori the $X_{ij}$ will not be disjoint. Therefore, we must modify the $X_{ij}$ so that each vertex is assigned to at most one $X_{ij}^+$ and at most one $X_{ij}^-$ . It turns out that this assignment problem is not too hard to incorporate into the flow problem described earlier (we simply add an edge of capacity $1$ from $v^+ \in V^+$ to $t_j$ if $v^+ \in X_{ij}^+$ and similarly if $v^- \in X_{ij}^-$ ). Solving the flow problem gives us the disjoint ‘extendable matchings’ we seek (meaning the $X_{ij}$ ’s are disjoint), which can greedily be extended to give the desired path system.

7.2. Preliminaries

We will require Vizing’s theorem for multigraphs in the proof of Lemma 7.8. Let H be an (undirected) multigraph (without loops). The multiplicity $\mu (H)$ of H is the maximum number of edges between two vertices of H, and, as usual, $\Delta (H)$ is the maximum degree of H. A proper k-edge-colouring of H is an assignment of k colours to the edges of H such that incident edges receive different colours.

Theorem 7.3 ([Reference Vizing34]; see e.g., [Reference Diestel5])

Any multigraph H has a proper k-edge colouring with $k = \Delta (H) + \mu (H)$ colours. In particular, by taking the largest colour class, there is a matching in H of size at least $e(H)/(\Delta (G)+\mu (G))$ .

We will also require the following simple proposition about decomposing acyclic digraphs into paths.

Proposition 7.4 [Reference Lo, Patel, Skokan and Talbot26, Proposition 2.6]

Let G be an acyclic digraph. Then the edges of G can be partitioned into $\sum _{v \in V(G)}\left | d^+(v) - d^-(v) \right |/2$ directed paths.

We will need the following property of regular digraphs; its proof is a simple exercise by considering $ \sum _{v \in V_{i*}} d^+(v)$ and $ \sum _{v \in V_{*i}} d^-(v)$ .

Proposition 7.5 [Reference Lo, Patel and Yıldız27, Proposition 3.2]

Let G be a d-regular digraph, $k\in \mathbb {N}$ , and $\mathcal {P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G)$ . Then, for all $i \in [k]$ , we have $d(|V_{i*}|-|V_{*i}|)= e( G_{i*} ) - e(G_{*i})$ .

The next proposition shows that we can consider a simpler subgraph G in $\mathcal {B}( G^0 , \mathcal {P} )$ .

Proposition 7.6. Let $G^0$ be a digraph, $k,d \in \mathbb {N}$ and $\mathcal {P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G^0)$ such that for all $i \in [k]$ , $e( G^0_{i*} ) - e(G^0_{*i}) = d ( |V_{i*}|-|V_{*i}| )$ . Then, by reordering $[k]$ if necessary, there exists a subdigraph G of $\mathcal {B}(G^0, \mathcal {P})$ such that

$$ \begin{align*} e(G) & \le d (k-1) \sum_{i \in [k]}\left| |V_{i*}|-|V_{*i}| \right|/2,\\ e( G_{i*} ) - e(G_{*i}) & = d(|V_{i*}|-|V_{*i}|) \text{ for all }i \in [k], \end{align*} $$

and for all $i,j \in [k]$ with $i \le j$ , $e(G_{ji}) = 0$ .

Proof. Define an auxiliary multidigraph $H^0$ on $[k]$ (with loops) such that there are precisely $e(G^0_{ij})$ many $ij$ edges in $H^0$ . There is a natural bijection between $E(G^0)$ and $E(H^0)$ . Let H be an acyclic subdigraph of $H^0$ obtained by successively removing all edges of a directed cycle where we treat loops also as cycles. Note that, for all $i \in [k]$ ,

(7.1) $$ \begin{align} d^+_{H}(i) - d^-_{H}(i) = d^+_{H^0}(i) - d^-_{H^0}(i) = e( G^0_{i*} ) - e(G^0_{*i}) = d(|V_{i*}|-|V_{*i}|). \end{align} $$

Since H is acyclic, by relabelling if necessary, we may assume that there are no edges $ji$ with $i \le j$ . By Proposition 7.4, $E(H)$ can be decomposed into $\sum _{i \in [k]}\left | d^+_{H}(i) - d^-_{H}(i) \right |/2$ directed paths. Since each path can have length at most $k-1$ (as H has k vertices), we have

The result follows by setting G to be the subdigraph of $G^0$ corresponding to H.

We need the following lemma from our previous work [Reference Lo, Patel and Yıldız27]. It states that given a set of matchings of low total maximum degree, one can select a relatively large number of the edges from each matching so that the union of selected edges is also a matching.

Lemma 7.7 [Reference Lo, Patel and Yıldız27, Lemma 4.2]

Let $k,\ell \in \mathbb {N}$ and $M_1,M_2,\ldots ,M_{\ell }$ be matchings with $\Delta \left (\bigcup _{i \in [\ell ]} M_i\right )\leq k$ . Suppose $e(M_i)> 8k^6\ln \ell $ for all $i \in [\ell ]$ . Then, there exists a matching $M \subseteq \bigcup _{i \in [\ell ]}M_i$ with $|M\cap M_i|\geq e(M_i)/2k^2$ for all $i \in [\ell ]$ .

Any graph (or digraph) either has a large matching or has all its edges incident to a small set of vertices (so these vertices have relatively large degree). The following lemma allows us to interpolate between these extremes and moreover does it simultaneously for all $G_{ij}$ . This corresponds to the ‘extendable matchings’ described in the sketch of proof.

Lemma 7.8. Let G be a digraph, $k,d\in \mathbb {N}$ , and $\mathcal {P} = \{V_{ij}:i,j\in [k]\}$ be a partition of $V(G)$ . Let $\theta \in (0,1)$ with $\theta d \ge 2$ . For all $i,j \in [k]$ and $v \in V(G)$ , let $X_{ij}^+ = \{v \in V(G) : d_{G_{ij}}^+(v) \ge \theta d \}$ and define $X_{ij}^-$ similarly. Then, for each $i,j \in [k]$ , there exists a matching $M_{ij}$ of $G_{ij}$ such that

1. (1) $ e(G_{ij}) \le \sum _{x^+ \in X_{ij}^+} d_{G_{ij}}^+(x^+) + \sum _{x^- \in X_{ij}^-} d_{G_{ij}}^-(x^-) + (6 k^2 \theta d) e( M_{ij} ) + 50 k^{10} \theta d $ ;

2. (2) for all $uv \in M_{ij}$ , $u \notin X_{ij}^+$ and $v \notin X_{ij}^-$ ;

3. (3) $\bigcup _{i,j \in [k]} M_{ij}$ is a matching.

Proof. Consider $i,j \in [k]$ . Let $H_{ij}$ be the multigraph obtained from $G_{ij}$ by deleting all the edges $uv$ with $u\in X_{ij}^+$ or $v \in X_{ij}^-$ , and by making all the edges undirected. Note that we have $\Delta (H_{ij}) + \mu (H_{ij}) \leq 2\theta d+2$ and

(7.2) $$ \begin{align} e(H_{ij}) \geq e(G_{ij}) - \left( \sum_{x^+ \in X_{ij}^+} d_{G_{ij}}^+(x^+) + \sum_{x^- \in X_{ij}^-} d_{G_{ij}}^-(x^-) \right). \end{align} $$

Then, by Vizing’s theorem for multigraphs (Theorem 7.3), there exists a matching $M_{ij}^H$ in $H_{ij}$ of size at least $e(H_{ij})/(2\theta d+2) \ge e(H_{ij})/3\theta d$ . Let $M_{ij}^0$ be the corresponding matching in $G_{ij}$ . If $e(M_{ij}^0) \le 16 k^{10}$ , then we set $M_{ij}^0$ to be empty. Thus, together with (7.2) we have

$$ \begin{align*} e(G_{ij}) \le \left( \sum_{x^+ \in X_{ij}^+} d_{G_{ij}}^+(x^+) + \sum_{x^- \in X_{ij}^-} d_{G_{ij}}^-(x^-) \right) + (3 \theta d) e( M_{ij}^0 ) + 50 k^{10} \theta d. \end{align*} $$

Observe that $\Delta \left (\bigcup _{i,j \in [k]} M_{ij}^0\right )\leq k$ . Apply Lemma 7.7 for nonempty matchings $M_{ij}^0$ (with $\ell \le k^2$ ) to obtain $M_{ij} \subseteq M_{ij}^0$ (set $M_{ij}=\emptyset $ if $M_{ij}^0=\emptyset $ ) such that $\bigcup _{i,j \in [k]}M_{ij}$ is a matching and, for all $i,j \in [k]$ , $e(M_{ij}) \ge e(M_{ij}^0) / 2k^2$ . The result follows.

Recall that for any directed matching M, $V^+(M)$ and $V^-(M)$ are the sets of starting and ending vertices of the directed edges in M, respectively. Formally, $V^+(M) =\{v\in V(M): vw\in E(M) \text { for some }w\in V(M) \}$ and similarly for $V^-(M)$ .

Proposition 7.9. Let G be a digraph, $k\in \mathbb {N}$ , and $\mathcal {P}=\{V_{ij}:i,j\in [k]\}$ be a partition of $V(G)$ . Let $W \subseteq V(G)$ . Suppose that, for each $i,j \in [k]$ with $i \ne j$ , there exist $Y_{ij}^+ \subseteq V_{i*}$ , $Y_{ij}^- \subseteq V_{*j}$ and $M_{ij} \subseteq G_{ij}$ such that

1. (a) $M = \bigcup _{i, j \in [k]} M_{ij}$ is a matching;

2. (b) both of $\{Y_{ij}^+, V^+(M_{ij}) : i,j \in [k] \}$ and $\{ Y_{ij}^-, V^-(M_{ij}) : i,j \in [k] \}$ are sets of disjoint sets;

3. (c) for all $y^+ \in Y_{ij}^+$ and $y^- \in Y_{ij}^-$ , $d_{G_{ij}}^+(y^+),d_{G_{ij}}^-(y^-) \ge 2\sum _{i,j \in [k]} ( |Y_{ij}^+| + |Y_{ij}^-| + e(M_{ij}) ) + |W|+1$ .

Then $\mathcal {B}(G, \mathcal {P})$ contains a path system $\mathcal {Q}$ such that the following hold:

1. (i) $e(\mathcal {Q}_{ij}) = |Y_{ij}^+| + |Y_{ij}^-| + e(M_{ij})$ for all $i,j \in [k]$ , where $\mathcal {Q}_{ij}=\mathcal {Q}\cap E(G_{ij})$ ;

2. (ii) $V(\mathcal {Q})\cap W \subseteq \bigcup _{i,j \in [k]} (Y_{ij}^+ \cup Y_{ij}^- \cup V(M_{ij})) $ ;

3. (iii) if $y^+\in Y_{ij}^+\setminus V^-(M)$ , then there exists $u\in V_{*j}\setminus W$ such that the single edge $y^+u$ is a path in $\mathcal {Q}$ , and a similar statement for $y^-\in Y_{ij}^-\setminus V^+(M)$ holds.

Proof. Start by setting $\mathcal {Q}=M$ and then for each $y \in Y_{ij}^+$ (and $y \in Y_{ij}^-$ ), greedily add an edge $yv$ (and $vy$ , respectively) in $G_{ij} \setminus W$ such that v avoids all current vertices in $\mathcal {Q}$ and all vertices in $\bigcup _{i,j}(Y_{ij}^+\cup Y_{ij}^-)$ , which is possible by (c). It is clear that we always maintain a path system, and that (i)–(iii) hold by construction.