Proof. Let $B_1$ and $B_2$ be disjoint subsets of vertices of size $\varepsilon n$ . In $D^*(n,C/n)$ , the probability that there is no edge between $B_1$ and $B_2$ is $(1-C/n)^{(\varepsilon n)^2}$ . Taking a union bound over all possible sets $B_1$ and $B_2$ of size exactly $\varepsilon n$ , we get that the probability that there is some disjoint $B_1$ and $B_2$ with no edge between $B_1$ and $B_2$ is at most

\begin{equation*} \binom {n}{\varepsilon n}^2 \left (1-\frac {C}{n}\right )^{(\varepsilon n)^2} \leq \exp\!\left ( 2 \varepsilon n \log \frac {e}{\varepsilon } - C \varepsilon ^2 n \right ) \leq \exp\! \left ( - \frac {C \varepsilon ^2 n}{2} \right ) .\end{equation*}

Proof. Let $P = (u_1, u_2, u_3, u_4)$ . Let $B_1 \;:\!=\; N^*(v)$ for $* = \sigma (u_1 u_2)$ , and let $B_2 \;:\!=\; N^*(v')$ for $* = \sigma (u_4 u_3)$ . Since $|B_1|, |B_2| \geq 2t + 1$ , there exists disjoint subsets $B'_{\!\!1} \subseteq B_1$ and $B'_{\!\!2} \subseteq B_2$ with $|B'_{\!\!1}|, |B'_{\!\!2}| \ge t$ which do not contain $v$ or $v'$ . Since $D$ is $t$ -bipseudorandom, there exists a bidirected edge $\overleftrightarrow{v_1 v_2}$ in $D$ with $v_1 \in B'_{\!\!1}$ and $v_2 \in B'_{\!\!2}$ . Then $(v, v_1, v_2, v')$ contains a copy of $P$ with startpoint $v$ and endpoint $v'$ .

Proof. Fix $P=(u_1, \dots, u_{2k+5})$ and an ordering $s_1, \dots, s_{k}$ of $S$ . We will find a $P$ -absorber for $(S,z)$ by applying Lemma 3.3 to various neighbourhoods of the $s_i$ , $v$ and $z$ . For each $i \in [k]$ , define

• $B_1 \;:\!=\; N^*(v)$ , where $* = \sigma (u_1 u_2)$ ,

• $B_{2i} \;:\!=\; N^*(s_i)$ , where $* = \sigma (u_{2i+2} u_{2i+1})$ ,

• $B_{2i+1} \;:\!=\; N^*(s_i)$ , where $* = \sigma (u_{2i+2} u_{2i+3})$ ,

• $B_{2k+2} \;:\!=\; N^*(z)$ , where $* = \sigma (u_{2k+5} u_{2k+4})$ .

Because of the minimum degree condition, $\vert B_i\vert \geq \alpha n$ , and therefore, we may take pairwise disjoint subsets $B'_{\!\!i}\subseteq B_i\setminus (U\cup S \cup \{z\})$ such that

\begin{equation*}\vert B'_{\!\!i}\vert \geq \frac {(\alpha n -\tfrac {\alpha }{2}n-k-1)}{2k+2} \geq \frac {\alpha n}{4(2k+2)} \geq 2\varepsilon n.\end{equation*}

Hence, an application of Lemma 3.3 yields a bidirected path $(v_1,\dots, v_{2k+2})$ such that $v_i\in B'_{\!\!i}$ for every $i\in [2k+2]$ .

Let $A \;:\!=\; \{v, v_1, \dots, v_{2k+2}\}$ . To see that $(A,v)$ is a $P$ -absorber for $(S,z)$ , observe that for every $s_i\in S$ , the pairs $(u_{2i+1},u_{2i+2})$ and $(u_{2i+2},u_{2i+3})$ in $P$ have the same directions of the underlying edges as the edges from the pairs $(v_{2i},s_i)$ and $(s_i,v_{2i+1})$ , allowing $s_i$ to play the role of $u_{2i+2}$ in $P$ ; see Figure 1. More precisely,

\begin{equation*} (v, v_1, \dots, v_{2i}, s_i, v_{2i+1}, \dots, v_{2k+2}, z) \end{equation*}

is a copy of $P$ in $D[A \cup \{s_i, z\}]$ with startpoint $v$ and endpoint $z$ .

Figure 1. An example of a $P$ -absorber $(A,v)$ for $(S,z)$ from Lemma 4.4 with $k=|S|=3$ . The double edges indicate that both orientations are present. For every $i\in \{1,2,3\}$ , $A \cup \{s_i, z\}$ contains a copy of $P$ in which vertex $v$ plays the role of $u_1$ , vertex $z$ plays the role of $u_{11}$ , vertex $s_i$ plays the role of $u_{2i+2}$ , and $v_j$ plays the role of $u_{j+1}$ for $j \leq 2i$ and $u_{j+2}$ for $j\gt 2i$ .

Proof. Let $X$ , $R$ , $v$ , and $v'$ be as in the statement of the lemma. Fix an arbitrary orientation of a path $P = (u_1, \dots, u_k)$ on $k \;:\!=\; |R| +\beta m$ vertices. Let $t \;:\!=\; 2\varepsilon n$ , and $X'\subseteq X$ be an arbitrary set of size $\beta m - 2t - 4 \geq \varepsilon n - 4 \gt 0$ . Let $R_0\;:\!=\;R\cup X'\setminus \{v,v'\}$ and $X_0\;:\!=\;X\setminus X'$ .

Observe that $D[R_0]$ is $\varepsilon n$ -bipseudorandom, so an application of Lemma 3.4 yields a bidirected path $Q_{\leftrightarrow }$ on exactly

(4.1) \begin{equation} |R_0| - 2 \varepsilon n = |R| -2 + |X'| - t = |R| +\beta m - 3t -6 \end{equation}

vertices. Denote by $w$ and $w'$ the startpoint and endpoint of $Q_\leftrightarrow$ , respectively.

Label $(R_0 \setminus V(Q_\leftrightarrow )) \cup \{v,w\}$ as $\{v_0 = v, v_1, \dots, v_t, v_{t+1}=w\}$ . For each $0 \leq i \leq t$ , we find a copy $(v_i, x_i, x'_{\!\!i}, v_{i+1})$ of the subpath $(u_{3i+1}, u_{3i+2}, u_{3i+3}, u_{3i+4})$ of $P$ , and we also find a copy $(w', x, x', v')$ of the subpath $(u_{k-3}, u_{k-2}, u_{k-1}, u_k)$ of $P$ , with $x_i, x'_{\!\!i}, x, x' \in X_0$ all distinct. This is possible by applying Proposition 4.3, observing that in total we will use $2(t+2)$ vertices of $X_0$ , and for any $U \subseteq X_0$ with $|U| \leq 2(t+2)$ and any $z, z' \in V(D) \setminus X_0$ , we have

\begin{equation*} \delta ^0(D[(X_0 \setminus U) \cup \{z,z'\}]) \geq 2\beta m - |X'| - |U| \geq \beta m \geq 2\varepsilon n + 1 .\end{equation*}

Recall that $\circ$ denotes concatenation. Thus $(v_0, x_0, x'_{\!\!0}, v_1, x_1, x'_{\!\!1}, v_2, \dots, v_{t+1}) \circ Q_\leftrightarrow \circ (w', x, x', v')$ contains a copy of $P$ with startpoint $v$ and endpoint $v'$ , since $3t+3 + |V(Q_\leftrightarrow )| + 3 = |R| + \beta m = k$ by (4.1); see Figure 2.

Proof. Given $\alpha$ and $\eta$ , define $p \;:\!=\;{\eta \alpha }/{2024}$ . Set $\beta \;:\!=\;\alpha/10$ and $\varepsilon \;:\!=\; p \beta/{6}$ . Let $P=(u_1, \dots, u_k)$ with $k\;:\!=\;\lceil \alpha n/4\rceil$ .

By applying the Chernoff bound for the hypergeometric distribution, we obtain a set $X \subseteq V(D)$ such that

1. (X1) $|X| = (1+\beta )m$ with $p n \lt m \lt |X| \lt 2p n$ and $\beta m \geq 5\varepsilon n +2$ , where we assume that $m$ and $\beta m$ are integers and ignore inconsequential rounding issues;

2. (X2) for every vertex $v \in V(D)$ and for every $* \in \{+,-\}$ , $|N^*(v) \cap X| \geq \frac{p}{2}|N^*(v)|\geq \frac{p\alpha n}{2} \geq 2\beta m +2$ .

Note that $X$ satisfies the hypothesis of Lemma 4.5, which will be used later. Arbitrarily choose two disjoint sets $Y, Z\subseteq V(D)\setminus X$ of sizes $2m$ and $3m$ , respectively. We form an auxiliary graph $H_m$ isomorphic to the graph from Lemma 4.6 on $X \cup Y \cup Z$ . We label $Z$ as $\{z_1, \dots, z_{3m}\}$ , and let $N_i \subseteq X \cup Y$ be a set of size exactly $40$ containing $N_{H_m}(z_i)$ .

We are now ready to construct the global absorber. We will identify a particular segment of $P$ for each $i \in [3m]$ , and use Lemma 4.4 to obtain a local absorber for $(N_i,z_i)$ for such a segment. These local absorbers combined will act as a global absorber since we can apply Lemma 4.5 to form the rest of $P$ with any appropriate set $R$ of vertices we wish to absorb, using exactly $\beta m$ vertices of $X$ in the process; the remaining part of $X$ along with $Y$ is matched to $Z$ via the property of $H_m$ given in Lemma 4.6, and this matching will tell us how to use each local absorber; see Figure 3.

Figure 2. A copy of $P$ in $D[R\cup X]$ with startpoint $v$ and endpoint $v'$ that covers all the vertices in $R$ , as found in Lemma 4.5.

Figure 3. The global absorber. The blue path with startpoint $v$ and endpoint $v'$ is a copy of $P$ covering $R$ , $X$ , $Y$ , and $Z$ . The red edges are the matching in the auxiliary graph $H_m$ , dictating which local absorber $A_i$ to use for each vertex in $X \cup Y$ .

Let $z_0 \in V(D) \setminus (X \cup Y \cup Z)$ be arbitrary. For each $i \in [3m]$ , we find a $(u_{84i-80}, \dots, u_{84i+4})$ -absorber $(A_i,z_{i-1})$ for $(N_i, z_i)$ such that $z_{i-1} \in A_i$ , the sets $A_i$ for $i\in [3m]$ are pairwise disjoint and disjoint from $X \cup Y \cup \{z_{3m}\}$ .Footnote ² This is possible by applying Lemma 4.4 (with $40$ playing the role of $k$ ) since we require the absorbers to be disjoint from at most

(4.2) \begin{equation} |A| = |X \cup Y| + 1 + \sum _{i=1}^{3m} |A_i| = 1 + (3+\beta )m + 3m\cdot 83 = (252+\beta )m+1 \leq \eta |V(P)| \leq \alpha n/2 \end{equation}

vertices, where we define

\begin{equation*} A \;:\!=\; X \cup Y \cup \{z_{3m}\} \cup \bigcup _{i \in [3m]} A_i .\end{equation*}

We claim that $A$ is a $P$ -global absorber, which by (4.2) has size at most $\eta |V(P)|$ .

Let $R \subseteq V(D) \setminus A$ be such that $|R| + |A| = |V(P)| = k$ , and let $v,v' \in R$ be distinct. By (X1) and (X2), we have that $\delta ^0(D[X \cup \{v, z_0\}]) \geq 2\beta m \geq 10\varepsilon n \geq 2 \varepsilon n +1$ , so we may apply Proposition 4.3 to obtain a copy $(v, x_1, x_2, z_0)$ of $(u_1, u_2, u_3, u_4)$ , where $x_1, x_2 \in X$ .

Let $\bar{X} \;:\!=\; X \setminus \{x_1, x_2\}$ , let $\bar{\beta } \;:\!=\; \beta - \frac{2}{m}$ , and let $\bar{R} \;:\!=\; (R \cup \{z_{3m}\}) \setminus \{v\}$ . By (X1), $|\bar{X}| = (1+\bar{\beta })m \geq m + 5\varepsilon n$ , and by (X2), for every $v \in V(D)$ and for every $* \in \{+,-\}$ , we have $|N^*(v) \cap \bar{X}| \geq 2\beta m$ . Therefore we may apply Lemma 4.5 to obtain a copy $Q$ of $(u_{252m+4}, \dots, u_k)$ in $D[\bar{X} \cup \bar{R}]$ covering $\bar{R}$ and exactly $\bar{\beta } m$ vertices of $\bar{X}$ with startpoint $z_{3m}$ and endpoint $v'$ .

Now we activate the local absorbers. Let $X' \;:\!=\; \bar{X} \setminus V(Q)$ , and note that $|X'| = m$ . By Lemma 4.6, there exists a matching between $Z$ and $X' \cup Y$ in $H_m$ . Fixing such a matching, let $z'_{\!\!i} \in N_i$ be the vertex matched to $z_i$ for each $i \in [3m]$ . Since $(A_i, z_{i-1})$ is a $(u_{84i-80}, \dots, u_{84i+4})$ -absorber for $(N_i, z_i)$ , there exists a copy $Q_i$ of $(u_{84i-80}, \dots, u_{84i+4})$ in $D[A_i \cup \{z_i, z'_{\!\!i}\}]$ with startpoint $z_{i-1}$ and endpoint $z_i$ . Concatenating as

\begin{equation*} (v, x_1, x_2, z_0) \circ Q_1 \circ \cdots \circ Q_{3m} \circ Q,\end{equation*}

we obtain a copy of $P$ in $D[A \cup R]$ with startpoint $v$ and endpoint $v'$ .

Proof of Theorem 1.3. Given $\alpha \gt 0$ , let $\varepsilon \gt 0$ be as in Lemma 4.7 on input $\alpha$ and $\eta \;:\!=\; 1/2$ . Set $C \;:\!=\; 4e/\varepsilon ^2$ . Let $D_0$ be an $n$ -vertex digraph with $\delta ^0(D_0)\geq \alpha n$ .

Given any orientation of a cycle $\mathcal C$ of length between $3$ and $n$ , our first aim is to prove that $D\;:\!=\;D_0\cup D^*(n, C/n)$ contains a copy of $\mathcal C$ with probability at least $1-e^{-n}$ . Note that Lemma 3.5 implies that $D^*(n, C/n)$ is $\varepsilon n$ -bipseudorandom with probability at least $1-e^{-n}$ ; thus, we may assume that $D$ is $\varepsilon n$ -bipseudorandom.

If $|\mathcal C| = 3$ , we use a proof similar to that of Proposition 4.3. Fix a vertex $v \in V(D)$ , and consider $N^+_D(v)$ and $N^-_D(v)$ , which both have size at least $\alpha n \geq 2\varepsilon n$ . Between these sets there is a bidirected edge, giving both possible orientations of $\mathcal C$ .

If $4 \leq \vert \mathcal C\vert \leq \alpha n/2$ , then we apply Lemma 3.4 to find a bidirected path $Q_\leftrightarrow$ in $D$ on $\vert \mathcal C\vert - 2$ vertices. Let $v$ and $v'$ be the startpoint and endpoint of $Q_\leftrightarrow$ , respectively, and let $P$ be a subpath of $\mathcal C$ on $3$ edges. Observe that $\delta ^0(D[(V(D)\setminus V(Q_\leftrightarrow )) \cup \{v,v'\}])\geq \alpha n/2 \geq 2\varepsilon n+1$ , and hence we may apply Proposition 4.3 to find a copy $Q$ of $P$ in $D$ with startpoint $v'$ and endpoint $v$ . Joining $Q$ and $Q_\leftrightarrow$ at both ends, we obtain a copy of $\mathcal C$ in $D$ .

If $|\mathcal C| \geq \alpha n/2$ , then let $P$ be a subpath of $\mathcal{C}$ on $\lceil{\alpha n/4}\rceil$ vertices. We apply Lemma 4.7 to find a $P$ -global absorber $A$ of size at most $\lceil{\alpha n/8}\rceil$ . Since $D[V(D)\setminus A]$ is $\varepsilon n$ -bipseudorandom, Lemma 3.4 yields a bidirected path on at least $n - |A| - 2\varepsilon n \gt n-|P|+2$ vertices disjoint from $A$ . Ignoring some vertices, let $Q_\leftrightarrow$ be a bidirected path on $\vert \mathcal C\vert -|P|+2$ vertices in $D[V(D) \setminus A]$ , and let $v$ and $v'$ be the startpoint and endpoint of $Q_\leftrightarrow$ , respectively. Let $R \subseteq (V(D) \setminus (V(Q_\leftrightarrow ) \cup A)) \cup \{v, v'\}$ with $v,v' \in R$ and $|R| = |P| - |A|$ . By Definition 4.1, there is a copy $Q$ of $P$ in $D$ with startpoint $v'$ and endpoint $v$ covering exactly the vertices of $R\cup A$ . Joining $Q$ and $Q_\leftrightarrow$ at both ends, we obtain a copy of $\mathcal C$ in $D$ .

Thus, for every orientation of a cycle $\mathcal C$ of length between $3$ and $n$ we have that $D_0 \cup D^*(n,C/n)$ contains a copy of $\mathcal C$ with probability at least $1-e^{-n}$ . By Lemma 3.1, $D_0 \cup D(n,C/n)$ contains a copy of $\mathcal C$ with probability at least $1-e^{-n}$ . Taking a union bound over all $n-2$ possible lengths and all at most $2^n$ possible orientations of each length, we have that $D_0 \cup D(n,C/n)$ contains every orientation of a cycle of length between $3$ and $n$ a.a.s.

Finally, to see that $D_0 \cup D(n,C/n)$ contains a cycle of length $2$ a.a.s., simply observe that $D_0$ has at least $\alpha n^2$ directed edges, and so the probability no edge of $D(n,C/n)$ is in the opposite orientation of an edge of $D_0$ is at most

\begin{equation*} (1-C/n)^{\alpha n^2} \leq e^{- C\alpha n}.\end{equation*}

Proof. Let $P = (u_1, \dots, u_k)$ . Label $S \cup \{z\}$ as $z_1, \dots, z_{\ell +1}$ , where $z_{\ell +1} \;:\!=\; z$ . We will find a $P$ -absorber for $(S,z)$ by applying Lemma 3.3 to various neighbourhoods of the $z_i$ , $v$ and $v'$ . Let $*_i \;:\!=\; +$ if $d^+(z_i) \geq \alpha n$ , and let $*_i \;:\!=\; -$ otherwise. Choose $\ell +1$ swap vertices of $P$ , $u_{i_1}, \dots, u_{i_{\ell +1}}$ , such that

• $i_{j+1} - i_j \geq 2$ for $0 \leq j \leq \ell +1$ , where $i_0 \;:\!=\; 2$ and $i_{\ell +2} \;:\!=\; k-1$ ,

• $i_{\ell +1} - i_{\ell } \geq 3$ ,

• $d_P^{*_j}(u_{i_j}) = 2$ for every $j \in [\ell +1]$ .

This is possible because $P$ has at least $3\ell +7$ swap vertices, and they alternate having in- or outdegree $2$ . Define

• $B_1\;:\!=\; N^*(v)$ , where $* = \sigma (u_1 u_2)$ ,

• $B_{i_j-2} \;:\!=\; B_{i_j-1} \;:\!=\; N^{*_j}(z_j)$ for $j \in [\ell ]$ ,

• $B_{i_{\ell +1}-3} \;:\!=\; B_{i_{\ell +1}-2} \;:\!=\; N^{*_{\ell +1}}(z_{\ell +1})$ ,

• $B_{k-4} \;:\!=\; N^*(v')$ , where $* = \sigma (u_k u_{k-1})$ ,

• $B_i \;:\!=\; V(D)$ for all remaining $i\in [k-4]$ .

Since $(v,v')$ is $\alpha$ -compatible with $P$ , and by the definition of the $*_j$ , we have that $|B_i| \geq \alpha n$ for every $i \in [k-4]$ . Since $|U| \leq \alpha n/2$ , there exists pairwise disjoint subsets $B'_{\!\!i} \subseteq B_i\setminus (U \cup S \cup \{z\})$ such that for all $i \in [k-4]$ ,

\begin{equation*}|B'_{\!\!i}| \geq \frac {(\alpha n- \alpha n/2 -\ell -1)}{(k-4)} \geq \frac {\alpha n}{4(k-4)} \geq 2 \varepsilon n.\end{equation*}

Lemma 3.3 gives a bidirected path $(v_1, \dots, v_{k-4})$ in $D$ with $v_i \in B' _i$ for every $i \in [k-4]$ . Let $A \;:\!=\; \{v, v_1, \dots, v_{k-4}, v'\}$ , and note that $A$ is disjoint from $U$ and $S \cup \{z\}$ .

To see that $A$ is a $P$ -absorber for $(S,z)$ , note that for every $z_j \in S$ , the path

\begin{equation*} (v, v_1, \dots, v_{i_j-2}, z_j, v_{i_j-1}, \dots, v_{i_{\ell +1}-3}, z, v_{i_{\ell +1}-2}, \dots, v_{k-4}, v') \end{equation*}

is a copy of $P$ in $D[A \cup \{z_j,z\}]$ with startpoint $v$ and endpoint $v'$ ; see Figure 4.

Proof. Let $U^* \;:\!=\; \{v \in V(D) \;:\; d^*(v) \geq \alpha n/2\}$ for each $* \in \{+,-\}$ . Since $\delta (D) \geq 2\alpha n$ , we have that

\begin{equation*} \frac {\alpha n}{2} (n-|U^*|) + n|U^*| \geq |E(D)| \geq \alpha n^2,\end{equation*}

which yields $|U^*| \geq \alpha n/2$ for each $* \in \{+,-\}$ . Since $\delta (D) \geq 2\alpha n$ , $U^+\cup U^-= V(D)$ .

If $|U^+ \setminus U^-| \geq \alpha n/2$ , then $V^+ \;:\!=\; U^+ \setminus U^-$ and $V^- \;:\!=\; U^-$ is a desired partition of $V(D)$ . Similarly, we get the desired partition if $|U^- \setminus U^+| \geq \alpha n/2$ .

Figure 4. An example of a $P$ -absorber $(A,v,v')$ for $(S=\{s_1,s_2\},z)$ from Lemma 6.4. The double edges indicate that both orientations are present. Notice that $u_4, u_7, u_{10}$ are swap vertices of $P$ , and for each fixed $i\in \{1,2\}$ , $A \cup \{s_i, z\}$ contains a copy of $P$ with startpoint vertex $v$ and endpoint $v'$ in which vertex $z$ plays the role of $u_{10}$ , and either $s_1$ plays the role of $u_4$ or $s_2$ plays the role of $u_7$ .

Otherwise, we must have that $|U^+ \cap U^-|\geq n-\alpha n \geq \alpha n +1$ . In this case, we partition $U^+ \cap U^-$ into $A \cup B$ with $||A| - |B|| \leq 1$ and set $V^+ \;:\!=\; (U^+ \setminus U^-) \cup A$ and $V^- \;:\!=\; (U^- \setminus U^+) \cup B$ .

Proof. By Fact 6.5, there exists a partition $V^+, V^-$ of $V(D)$ such that for each $* \in \{+,-\}$ we have that $|V^*| \geq \alpha n/2$ and $d^*(v) \geq \alpha n/2$ for every $v \in V^*$ .

Let $X$ be a randomly selected subset of $V(D)$ of size $(1+\beta )m$ . Set $X^+ \;:\!=\; V^+\cap X$ and $ X^-\;:\!=\; V^-\cap X$ . Then by the Chernoff bound for the hypergeometric distribution, with positive probability the following hold: for every $v \in V(D)$ and for each $* \in \{+,-\}$ , $d^*(v) \geq \alpha n/2$ implies $|N^*(v) \cap X| \geq \alpha m/ 3 \geq 2 \beta m$ ; $|X^+ |, |X^-| \geq 2 \beta m$ . Thus, $X$ is an $(\alpha,\beta,m)$ -reservoir, as desired.

Proof. Let $X^+$ , $X^-$ be the partition of $X$ as given in Definition 6.6. Let $P = (u_1, \dots, u_k)$ . Fix an arbitrary $U \subseteq X$ with $|U| + k \leq \frac{3}{2} \beta m$ , and let $v_1, v_k \in V(D) \setminus X$ be such that $(v_1, v_k)$ is $(\alpha/2)$ -compatible with $P$ . We will construct a copy $Q = (v_1, \dots, v_k)$ of $P$ in $D[(X \setminus U) \cup \{v_1, v_k\}]$ in stages, in all but the final step adding two vertices at a time.

For some $i \leq (k-2)/2$ , assume that there is a copy $Q^{\leq 2i-1} = (v_1, \dots, v_{2i-1})$ of $(u_1, \dots, u_{2i-1})$ in $D[(X\setminus U) \cup \{v_1\}]$ such that $d^*(v_{2i-1}) \geq \alpha n/2$ , where $* = \sigma (u_{2i-1} u_{2i})$ . Note that $Q^{\leq 1}\;:\!=\; (v_1)$ satisfies this for $i=1$ . Let $B_1 \;:\!=\; N^*(v_{2i-1}) \cap X$ for $* = \sigma (u_{2i-1} u_{2i})$ and $B_2 \;:\!=\; X^*$ for $* = \sigma (u_{2i+1} u_{2i+2})$ . Since $|B_1|, |B_2| \geq 2\beta m$ and $|U \cup V(Q^{\leq 2i-1})| \leq |U| + k \leq \frac{3}{2} \beta m$ , there exist disjoint subsets $B'_{\!\!i} \subseteq B_i$ of size at least $\beta m/4 \geq \varepsilon n$ disjoint from $U$ and $V(Q^{\leq 2i-1})$ . Since $D$ is $\varepsilon n$ -bipseudorandom, there exists $v_{2i} \in B'_{\!\!1}$ and $v_{2i+1} \in B'_{\!\!2}$ such that $(v_{2i-1}, v_{2i}, v_{2i+1})$ is a copy of $(u_{2i-1}, u_{2i}, u_{2i+1})$ . We thus obtain $Q^{\leq 2i+1} \;:\!=\; Q^{\leq 2i-1} \circ (v_{2i-1}, v_{2i}, v_{2i+1})$ as a copy of $(u_1, \dots, u_{2i+1})$ in $D[(X \setminus U) \cup \{v_1\}]$ with $d^*(v_{2i+1}) \geq \alpha n/2$ for $* = \sigma (u_{2i+1} u_{2i+2})$ .

If $k$ is even, then we slightly modify the last step, after constructing $Q^{\leq k-3}$ . Similarly as before, we can find a bidirected edge $\overleftrightarrow{v_{k-2} v_{k-1}}$ between $N^*(v_{k-3}) \cap X$ with $* = \sigma (u_{k-3} u_{k-2})$ and $N^*(v_k) \cap X$ with $* = \sigma (u_k u_{k-1})$ disjoint from $U \cup V(Q^{\leq k-3})$ . Thus $Q \;:\!=\; Q^{\leq k-3} \circ (v_{k-3}, v_{k-2}, v_{k-1}, v_k)$ contains a copy of $P$ with startpoint $v_1$ , endpoint $v_k$ , and all internal vertices in $X \setminus U$ .

If $k$ is odd, then we construct $Q^{\leq k-4}$ and use Lemma 3.3 in place of the pseudorandom property. Let $B_1 \;:\!=\; N^*(v_{k-4}) \cap X$ for $* = \sigma (u_{k-4} u_{k-3})$ , $B_2\;:\!=\; X$ , and $B_3\;:\!=\; N^*(v_k) \cap X$ with $* = \sigma (u_k u_{k-1})$ . Since $|B_1|, |B_2|, |B_3| \geq 2\beta m$ and $|U \cup V(Q^{\leq k-4})| \leq \frac{3}{2} \beta m$ , there exists disjoint subsets $B'_{\!\!i} \subseteq B_i$ of size at least $\beta m/6 \geq 2\varepsilon n$ disjoint from $U$ and $V(Q^{\leq k-4})$ . By Lemma 3.3, there exists a bidirected path $(v_{k-3}, v_{k-2}, v_{k-1})$ with $v_{k-3} \in B_1$ , $v_{k-2} \in B_2$ , and $v_{k-1} \in B_3$ . Thus $Q \;:\!=\; Q^{\leq k-4} \circ (v_{k-4}, v_{k-3}, v_{k-2}, v_{k-1}, v_k)$ contains a copy of $P$ with startpoint $v_1$ , endpoint $v_k$ , and all internal vertices in $X \setminus U$ .

Proof. Label $R$ as $\{v_1, \dots, v_\ell \}$ , with $v \;=\!:\; v_1$ and $v' \;=\!:\; v_\ell$ . Set $k \;:\!=\; \beta m + \ell$ and write $P = (u_1, \dots, u_k)$ . So $P$ contains at least $4\ell -6$ swap vertices and $(v_1, v_\ell )$ is $(\alpha/2)$ -compatible with $P$ .

Choose $\ell -2$ swap vertices of $P$ , $u_{i_2}, \dots, u_{i_{\ell -1}}$ , such that

• $i_{j+1} - i_j \geq 3$ for $j \in [\ell -1]$ , where $i_1 \;:\!=\; 1$ and $i_{\ell }\;:\!=\; k$ ,

• $d^*(v_j) \geq \alpha n$ with $* = \sigma (u_{i_j} u_{i_j+1})$ , for every $2 \leq j \leq \ell -1$ .

This is possible because $P$ has at least $4\ell -6$ swap vertices, and they alternate having in- or outdegree $2$ . Let $P_j \;:\!=\; (u_{i_j}, \dots, u_{i_{j+1}})$ for $j \in [\ell -1]$ , and observe that $(v_j, v_{j+1})$ is $(\alpha/2)$ -compatible with $P_j$ . Since the number of vertices used in total is at most $|P| \leq \frac{3}{2} \beta m$ , by Lemma 6.8, we can find pairwise internally disjoint copies $Q_j$ of $P_j$ in $D[X \cup \{v_j, v_{j+1}\}]$ with startpoint $v_j$ and endpoint $v_{j+1}$ . Concatenating the $Q_j$ as $Q \;:\!=\; Q_1 \circ \cdots \circ Q_{\ell -1}$ , we obtain a copy of $P$ in $D[R \cup X]$ with startpoint $v_1$ and endpoint $v_{\ell }$ covering $R$ ; see Figure 5.

Proof. Fix $k \leq (1-8\varepsilon )n - |U|$ , $*_1$ , and $*_2$ . By Fact 6.5, and as $|U| \leq \alpha n/4$ , we can partition $V(D) \setminus U$ as $V^+ \cup V^-$ , where for each $* \in \{+,-\}$ we have that $|V^*| \geq \alpha n/4$ , and $d^*_D(v) \geq \alpha n/2$ for every $v \in V^*$ . Lemma 3.4 gives a bidirected path $Q^*$ in $D[V^*]$ on at least $|V^*| - 2\varepsilon n$ vertices for each $* \in \{+,-\}$ .

Case 1: $*_1 \neq *_2$ . We take the last $\varepsilon n$ vertices of $Q^-$ and the first $\varepsilon n$ vertices of $Q^+$ and find a bidirected edge between them, which exists since $D$ is $\varepsilon n$ -bipseudorandom. This gives a bidirected path $Q$ on at least $|V(Q^-)| + |V(Q^+)| - 2 \varepsilon n \geq n - |U| -6\varepsilon n$ vertices. Truncating $Q$ at both ends, we obtain a bidirected path on $k$ vertices with startpoint in $V^{*_1}$ and endpoint in $V^{*_2}$ .