Bounds on the Lengths of Multiple Short Directed Cycles

1. Introduction

Throughout this paper, all digraphs are non-empty and simple (i.e., contain no parallel arcs or loops). We denote the set of positive integers by ${\mathbb{N}}^{*}$. Let G be a digraph with vertex set $V\left(G\right)$ and arc set $A\left(G\right)$. The order of G is $\left|G\right|=\left|V\right(G\left)\right|$. An arc $(x,y)\in A\left(G\right)$ makes y an out-neighbor of x and x an in-neighbor of y. The outdegree of a vertex x, denoted $d^{+}\left(x\right)$, is the number of its out-neighbors; similarly, its indegree $d^{-}\left(x\right)$ is the number of its in-neighbors. A vertex z is a second out-neighbor of x if there exists a vertex y such that $(x,y)\in A\left(G\right),(y,z)\in A\left(G\right),(x,z)\notin A\left(G\right),\mathrm{and}x,y,z\mathrm{are}\mathrm{pairwise}\mathrm{distinct}.$ The second outdegree of x is the number of its second out-neighbors. The minimum outdegree and minimum indegree of G are defined by ${\delta }^{+}\left(G\right)=min\{d^{+}\left(x\right);x\in V\left(G\right)\}$ and ${\delta }^{-}\left(G\right)=min\{d^{-}\left(x\right);x\in V\left(G\right)\}$. A directed cycle C of length n in G is a subdigraph with vertex set $\{x_1,x_2,\dots ,x_n\}$ and arc set $\{(x_1,x_2),(x_2,x_3),\dots ,(x_{n-1},x_n),(x_n,x_1)\},$ where all $x_i$ are distinct. Its length (equivalently its number of vertices) is n.

In this paper, we shift focus from finding a single short directed cycle to guaranteeing the existence of multiple “short” directed cycles. Specifically, we investigate conditions under which a digraph with given minimum outdegree contains m distinct or vertex-disjoint directed cycles of bounded length.

The novelty of our approach lies in shifting the paradigm of the Caccetta–Häggkvist Conjecture from a single-cycle search to a multi-cycle packing problem. While one might intuitively expect that forcing multiple bounded-length cycles requires a strictly stronger outdegree condition, a highly surprising and significant aspect of our results is that the density condition on the minimum outdegree remains identical to that of finding a single cycle. By extending Shen’s theorem stated formally in Section 2, we demonstrate that a digraph with a sufficiently large order structurally accommodates not just one, but m short cycles under the exact same outdegree constraints. This reveals a deeper structural richness in digraphs with prescribed outdegrees and provides a new quantitative tool that could be instrumental in resolving related, broader conjectures. Our main results are stated below and proved in Section 3.

Main Theorem 1.Let $m\in {\mathbb{N}}^{*}$, $\alpha \in (0,1)$ and $n_0={\displaystyle \frac{(m-1)(1+74\alpha )}{{\alpha }^3}}$. Then, every digraph G on $n\ge n_0$ vertices with minimum outdegree $\alpha n$ has m vertex-disjoint directed cycles, each with a length of at most ${\displaystyle \frac{1}{\alpha }}+74$.

Main Theorem 2.Let $m\in {\mathbb{N}}^{*}$, $\alpha \in (0,1)$ and $n_0={\displaystyle \frac{m-1}{{\alpha }^2}}$. Then, every digraph G on $n\ge n_0$ vertices with minimum outdegree $\alpha n$ has m distinct directed cycles, each with a length of at most ${\displaystyle \frac{1}{\alpha }}+74$.

2. Literature Review

The study of cycles forms a foundational pillar of graph theory [1], with extensive literature dedicated to the classical undirected case [2]. In undirected graphs, algorithmic approaches and structural constraints, often leveraging invariants such as the clique number or circumference, have been widely employed to study cycle existence and cycle lengths [3]. However, extending these results to directed graphs introduces significant complexity, as many structural properties available in undirected graphs no longer hold in the directed setting, requiring substantially different combinatorial and algorithmic techniques [4].

Regarding cycles in digraphs, Camion [5] proved that every strongly connected tournament contains a Hamiltonian directed cycle. More generally, Bondy [6] showed that every strongly connected n-chromatic digraph contains a directed cycle of length at least n. Recently, Al-Mniny et al. [7] established a special case of Cohen et al.’s conjecture on oriented cycles in strongly connected chromatic digraphs [8] by studying oriented cycles consisting of four blocks. For cycles with an arbitrary number of blocks, El-Joubbeh et al. [9] obtained linear bounds in Hamiltonian digraphs. These bounds were subsequently improved in [10,11,12] using secant edges [13].

Within the study of cycle packing in digraphs, Moon [14] proved that every strongly connected tournament T is pancyclic, that is, it contains cycles of all lengths $3,4,\dots ,n$, where n is the number of vertices of T. Reed et al. [15] proved that for every integer $n\ge 0$, there is an integer $t\ge 0$ such that for every digraph G, either G has n vertex-disjoint directed cycles, or G can be made acyclic by deleting at most t vertices. In addition, Bessy et al. [16] proved that every strongly connected digraph can be covered by at most $\alpha \left(D\right)$ directed cycles, where $\alpha \left(D\right)$ denotes the independence number of D.

A central problem in this domain is determining the minimum outdegree required to force a specific number of vertex-disjoint directed cycles. Bermond and Thomassen [17] conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains k vertex-disjoint directed cycles. While this conjecture remains open in full generality, Alon [18] proved a linear upper bound, showing that a minimum outdegree of $64k$ suffices. This bound has been subsequently improved and studied in various partial results. Our main theorems complement this body of literature by approaching the packing problem from a different angle: rather than scaling the outdegree condition proportionally with the number of desired cycles m, we fix the single-cycle outdegree density $\alpha n$ and leverage a sufficiently large graph order to force m bounded-length cycles.

A prominent open problem concerning short directed cycles is the Caccetta–Häggkvist Conjecture:

Conjecture 1 

([19]). Every digraph on n vertices with minimum outdegree of at least k contains a directed cycle of length at most $⌈{\displaystyle \frac{n}{k}}⌉$.

One strategy toward resolving this conjecture is to establish the existence of a directed cycle of length at most ${\displaystyle \frac{n}{k}}+c$ for some absolute constant c. Chvátal and Szemerédi [20] established Conjecture 1 with $c=2500$, Nishimura [21] later improved their proof to reduce the bound to $c=304$, and Shen subsequently proved Conjecture 1 under the sharper condition $n\ge 2k^2-3k+1$ [22], while in the general case, further sharpening the earlier bounds to show that $c=73$ suffices.

Theorem 1 

([23]). Every digraph on n vertices with minimum outdegree at least k contains a directed cycle of length at most ${\displaystyle \frac{n}{k}}+73$.

Lichiardopol [24] proved Conjecture 1 when the independence number is two. Then Hompe [25] generalized the previous result, proving Conjecture 1 when the independence number is at most $(k+1)/2$. Despite these advances, the conjecture remains open even in the special case when the minimum outdegree and the minimum indegree are both at least k, where$k={\displaystyle \frac{n}{3}}$. For a vertex x in a digraph G, let $d^{+}\left(x\right)$ and $d^{++}\left(x\right)$ denote the number of vertices at distance one and two from x, respectively. In this setting, as part of an effort to establish this special case, Seymour proposed the following conjecture, which clearly encompasses this case.

Conjecture 2 

([26]). Every simple digraph has a vertex x with $d^{+}\left(x\right)\le d^{++}\left(x\right)$.

Numerous partial results address Seymour’s conjecture (see, e.g., [27,28,29,30]), yet the full conjecture remains unresolved. One of the partial results proved by Havet et al. [31] is that every tournament that has a positive minimum outdegree has two vertices satisfying Conjecture 2. This fact was used by Fidler et al. [32] to prove Conjecture 2 for digraphs missing a matching. In fact, Fidler et al. assigned some convenient orientations to all the missing edges of the digraph D. The obtained tournament T has a positive minimum outdegree, and thus based on [31] it has two vertices satisfying Seymour’s conjecture. Then, Fidler showed that one of these two vertices satisfies the conjecture in the initial digraph D. These developments suggest that guaranteeing multiple occurrences of certain graph elements or structures may provide new insights for resolving problems.

In this paper, we strictly limit our scope and motivation to the Caccetta–Häggkvist paradigm, exploring how minimum outdegree conditions can guarantee the multi-cycle packing of short directed cycles.

3. Proofs of the Main Theorems

Proof 

(Proof of Main Theorem 1). We first establish a parameterized bound. Let $c>1$ and set $n_0^{\prime }={\displaystyle \frac{(m-1)(c-\alpha )(c+73\alpha )}{(c-1){\alpha }^2}}$. We will show by induction on m that every digraph G on $n\ge n_0^{\prime }$ vertices with minimum outdegree $\alpha n$ has m vertex-disjoint directed cycles, each with a length of at most ${\displaystyle \frac{c}{\alpha }}+73$. Consider a digraph G on $n\ge n_0^{\prime }$ vertices with minimum outdegree $\alpha n$. Let $k=⌈\alpha n⌉$ denote the ceiling of $\alpha n$. The proof is by induction on m.

Suppose that $m=1$. Theorem 1 guarantees the existence of a cycle C of length $\left|C\right|\le {\displaystyle \frac{n}{k}}+73\le {\displaystyle \frac{1}{\alpha }}+73\le {\displaystyle \frac{c}{\alpha }}+73$. So it is true for $m=1$.

Suppose that it is true for $m-1$. By the induction hypothesis, there are $m-1$ disjoint cycles $C_1,\dots ,C_{m-1}$ each of length at most ${\displaystyle \frac{c}{\alpha }}+73$. Let $G^{\prime }$ denote the graph obtained from G by deleting all the vertices of the cycles $C_1,\dots ,C_{m-1}$. Let $n^{\prime }$ be the number of vertices of $G^{\prime }$ and $k^{\prime }$ be the minimum outdegree of $G^{\prime }$.

Before applying Shen’s theorem to find the next cycle, we must rigorously verify that the reduced digraph $G^{\prime }$ satisfies the necessary assumptions—specifically, that its minimum outdegree $k^{\prime }$ is strictly positive. The number of remaining vertices is $n^{\prime }=n-{\sum }_{i=1}^{m-1}\left|C_i\right|$. Because we deleted exactly ${\sum }_{i=1}^{m-1}\left|C_i\right|$ vertices in total from G, the outdegree of any remaining vertex decreases by at most this amount. Consequently, the new minimum outdegree satisfies $k^{\prime }\ge \alpha n-{\sum }_{i=1}^{m-1}\left|C_i\right|$.

To show that $k^{\prime }>0$, we establish a stronger inequality. By the induction hypothesis, $|C_i|\le {\displaystyle \frac{c}{\alpha }}+73$ for all $1\le i\le m-1$. Since $n\ge n_0^{\prime }$, we have

$$\begin{array}{cc}\hfill (\alpha n-{\displaystyle \sum _{i=1}^{m-1}}|C_i\left|\right)-{\displaystyle \frac{\alpha }{c}}(n-{\displaystyle \sum _{i=1}^{m-1}}|C_i\left|\right)& ={\displaystyle \frac{\alpha (c-1)}{c}}n-{\displaystyle \frac{(c-\alpha )}{c}}{\displaystyle \sum _{i=1}^{m-1}}\left|C_i\right|\hfill \\ \hfill & \ge {\displaystyle \frac{\alpha (c-1)}{c}}n-{\displaystyle \frac{(c-\alpha )(m-1)(c+73\alpha )}{c\alpha }}\hfill \\ \hfill & \ge 0\hfill \end{array}$$

Hence, $k^{\prime }\ge \alpha n-{\sum }_{i=1}^{m-1}|C_i|\ge {\displaystyle \frac{\alpha }{c}}(n-{\sum }_{i=1}^{m-1}|C_i\left|\right)={\displaystyle \frac{\alpha }{c}}{n}^{\prime }$. Since $n\ge n_0^{\prime }$ guarantees $n^{\prime }>0$ and we have ${\displaystyle \frac{\alpha }{c}}>0$, it follows strictly that $k^{\prime }>0$. Having verified that $G^{\prime }$ possesses a positive minimum outdegree, we can now apply Theorem 1 to $G^{\prime }$. This yields a cycle $C_m$ in $G^{\prime }$ of the following length:

$$\begin{array}{cc}\hfill |C_m|\le {\displaystyle \frac{n^{\prime }}{k^{\prime }}}+73& \le {\displaystyle \frac{n^{\prime }}{\frac{\alpha }{c}{n}^{\prime }}}+73\hfill \\ \hfill & \le {\displaystyle \frac{c}{\alpha }}+73\hfill \end{array}$$

To obtain the specific bound stated in the theorem, we take $c=1+\alpha$. For this choice of c, the required minimum number of vertices evaluates precisely to $n_0^{\prime }={\displaystyle \frac{(m-1)\left(1\right)(1+74\alpha )}{\alpha ·{\alpha }^2}}={\displaystyle \frac{(m-1)(1+74\alpha )}{{\alpha }^3}}=n_0$. Thus, for $n\ge n_0$, the digraph G has m vertex-disjoint directed cycles each of length at most ${\displaystyle \frac{1+\alpha }{\alpha }}+73={\displaystyle \frac{1}{\alpha }}+74$. This completes the proof. □

Proof 

(Proof of Main Theorem 2). Similarly, we first establish a parameterized bound. Let $c>1$ and set $n_0^{\prime }={\displaystyle \frac{(m-1)(c-\alpha )}{(c-1)\alpha }}$. We will show by induction on m that every digraph G on $n\ge n_0^{\prime }$ vertices with minimum outdegree $\alpha n$ has m distinct directed cycles, each with a length of at most ${\displaystyle \frac{c}{\alpha }}+73$.

Consider a digraph G on $n\ge n_0^{\prime }$ vertices with minimum outdegree $\alpha n$. Let $k=⌈\alpha n⌉$ denote the ceiling of $\alpha n$. The proof is by induction on m.

Suppose that $m=1$. Theorem 1 guarantees the existence of a cycle C of length $\left|C\right|\le {\displaystyle \frac{n}{k}}+73\le {\displaystyle \frac{1}{\alpha }}+73\le {\displaystyle \frac{c}{\alpha }}+73$. So, it is true for $m=1$.

Suppose that it is true for $m-1$. By the induction hypothesis, there are $m-1$ distinct directed cycles $C_1,\dots ,C_{m-1}$, each of length at most ${\displaystyle \frac{c}{\alpha }}+73$. Let A be a set of vertices obtained by choosing one vertex from each cycle $C_i$. Let r denote the cardinality of A. Let $G^{\prime }$ denote the graph obtained from G by deleting the vertices in A. Let $n^{\prime }$ be the number of vertices of $G^{\prime }$ and $k^{\prime }$ be the minimum outdegree of $G^{\prime }$.

As before, we must verify that $G^{\prime }$ satisfies the assumptions of Shen’s theorem by proving $k^{\prime }>0$. The number of vertices in $G^{\prime }$ is $n^{\prime }=n-r$. Since we deleted exactly r vertices, the outdegree of any remaining vertex drops by at most r, yielding $k^{\prime }\ge \alpha n-r$. Note that $r\le m-1$ because we selected exactly one vertex from each of the $m-1$ cycles (and some selections may overlap), which establishes the specific inequalities $k^{\prime }\ge \alpha n-r\ge \alpha n-m+1$. Since $n\ge n_0^{\prime }$, we evaluate the following inequality:

$$\begin{array}{cc}\hfill (\alpha n-r)-{\displaystyle \frac{\alpha }{c}}(n-r)& ={\displaystyle \frac{\alpha (c-1)}{c}}n-{\displaystyle \frac{(c-\alpha )}{c}}r\hfill \\ \hfill & \ge {\displaystyle \frac{\alpha (c-1)}{c}}n-{\displaystyle \frac{(c-\alpha )}{c}}(m-1)\hfill \\ \hfill & \ge 0\hfill \end{array}$$

Hence, $k^{\prime }\ge \alpha n-r\ge {\displaystyle \frac{\alpha }{c}}(n-r)={\displaystyle \frac{\alpha }{c}}{n}^{\prime }$. Because $n\ge n_0^{\prime }$ guarantees $n^{\prime }>0$ and ${\displaystyle \frac{\alpha }{c}}>0$, this definitively confirms $k^{\prime }>0$. Now that the strictly positive minimum outdegree is verified, we can safely apply Theorem 1 to $G^{\prime }$. This guarantees that $G^{\prime }$ has a cycle $C_m$ of the following length:

$$\begin{array}{cc}\hfill |C_m|\le {\displaystyle \frac{n^{\prime }}{k^{\prime }}}+73& \le {\displaystyle \frac{n^{\prime }}{\frac{\alpha }{c}{n}^{\prime }}}+73\hfill \\ \hfill & \le {\displaystyle \frac{c}{\alpha }}+73\hfill \end{array}$$

To obtain the specific bound stated in the theorem, we take $c=1+\alpha$. For this choice of c, the required minimum number of vertices evaluates precisely to $n_0^{\prime }={\displaystyle \frac{(m-1)\left(1\right)}{\alpha ·\alpha }}={\displaystyle \frac{m-1}{{\alpha }^2}}=n_0$. Thus, for $n\ge n_0$, the digraph G has m distinct directed cycles, each of length of at most ${\displaystyle \frac{1+\alpha }{\alpha }}+73={\displaystyle \frac{1}{\alpha }}+74$. This completes the proof. □

4. Discussion and Conclusions

To intuitively illustrate the vertex-deletion procedure used in our induction proofs, consider the case of finding $m=2$ cycles. After the first application of Shen’s theorem yields a short cycle, $C_1$, the host graph G must be reduced to find a second cycle, $C_2$. If our goal is to find vertex-disjoint cycles (as in Main Theorem 1), we delete the entire vertex set of $C_1$. By removing $V\left(C_1\right)$, we guarantee that the new cycle $C_2$ shares absolutely no vertices with $C_1$. Conversely, to find merely distinct cycles (as in Main Theorem 2), deleting the entirety of $C_1$ is unnecessarily destructive to the outdegrees of the remaining vertices. Instead, we delete just a single vertex $v\in V\left(C_1\right)$. Because the new cycle $C_2$ is found in $G-\left\{v\right\}$, it strictly avoids v, making it structurally impossible for $C_2$ to be identical to $C_1$. This guarantees the cycles are distinct while preserving much more of the original graph’s density.

In this paper, we have established a bound on the lengths of multiple distinct and vertex-disjoint directed cycles in digraphs with prescribed minimum outdegree conditions. Our main results show that for any $m\in {\mathbb{N}}^{*}$ and $\alpha \in (0,1)$, every digraph G on sufficiently many vertices with minimum outdegree at least $\alpha n$ contains m distinct or vertex-disjoint directed cycles, each with a length of at most ${\displaystyle \frac{1}{\alpha }}+74$. This extends the classical results of Shen on single short cycles to the setting of multiple cycles, providing an explicit quantitative relationship between the minimum outdegree density and the existence of bounded-length cycle structures.

Regarding the sharpness and optimality of our bounds, the obtained cycle length of ${\displaystyle \frac{1}{\alpha }}+74$ is closely tied to the additive constant in Shen’s theorem. Our bound exceeds Shen’s best-known bound for a single cycle (${\displaystyle \frac{1}{\alpha }}+73$) by exactly one, which is a byproduct of the density reduction inherent in our recursive vertex-deletion method. However, this bound is not strictly optimal; the Caccetta–Häggkvist Conjecture postulates an optimal bound of $\lceil {\displaystyle \frac{1}{\alpha }}\rceil$ for a single cycle. Furthermore, a limitation of our approach lies in the required minimum order of the host graph, $n_0$. While $n_0$ grows linearly with the number of desired cycles m, it grows polynomially as the density $\alpha$ becomes small (specifically, proportional to ${\displaystyle \frac{1}{{\alpha }^2}}$ for distinct cycles and ${\displaystyle \frac{1}{{\alpha }^3}}$ for vertex-disjoint cycles). Reducing this required minimum order remains a significant challenge for future work.

Several natural questions arise from our work. First, it would be interesting to determine whether the bound ${\displaystyle \frac{1}{\alpha }}+74$ on cycle length can be improved, particularly in relation to the conjectured optimal bound. Furthermore, while this work is inherently theoretical, cycle packing and the bounding of short directed cycles have natural applications in computer science, particularly in algorithmic deadlock detection within distributed systems and routing protocols in directional communication networks. Finally, it remains an open question whether similar bounds can be established for other cycle-related structures, such as cycles satisfying additional constraints (e.g., Hamiltonian properties) or cycles with prescribed intersections. We hope that our methods and results contribute to a deeper understanding of cycle structures in digraphs with minimum degree conditions and stimulate further research in this active area of graph theory.