Dijoin Conjecture in Digraph Theory

• The Dijoin Conjecture is a key concept in digraph theory, defined in both dicut-dijoin packing and inversion-number settings.
• In the packing context, it posits that the minimum size of a dicut equals the maximum number of disjoint dijoins, influencing connectivity and decycling strategies.
• In the inversion-number framework, it conjectures that the inversion number is additive under graph joins, with proven cases and notable counterexamples guiding current research.

The term Dijoin Conjecture is used in two distinct digraph-theoretic settings. In the classical packing setting, a dijoin is a subset of arcs that intersects every dicut, and Woodall conjectured in 1976 that in every digraph the minimum size of a dicut equals the maximum number of disjoint dijoins (Cornuéjols et al., 2023). In the inversion-number setting, Bang-Jensen, da Silva, and Havet conjectured that for oriented graphs $L$ and $R$,

$$\mathrm{inv}(L \rightarrow R)=\mathrm{inv}(L)+\mathrm{inv}(R),$$

where $L \rightarrow R$ is the oriented join obtained by adding all arcs from $L$ to $R$ (Alon et al., 2022). The two statements concern different objects—arc sets meeting directed cuts versus additivity of inversion number under a graph operation—but both have become focal problems in the study of directed cuts, strong connectivity, and decycling in oriented graphs (Behague et al., 12 Sep 2025).

1. Definitions and basic frameworks

In the packing literature, for a digraph $D=(V,A)$, a dicut is a cut $\delta^+(U)\subseteq A$ for some nonempty proper subset $U\subsetneq V$ such that $\delta^-(U)=\emptyset$. A dijoin is a subset $R$0 that intersects every dicut, and more generally a $R$1-dijoin intersects every dicut at least $R$2 times (Cornuéjols et al., 2023). The minimum size of a dicut is commonly denoted by $R$3, and in the weighted setting one considers the minimum dicut weight and packings of dijoins subject to arc capacities $R$4 (Abdi et al., 2022).

In the inversion-number literature, for an oriented graph $R$5 and a subset $R$6, the inversion of $R$7 is obtained by reversing all arcs with both endpoints in $R$8. The inversion number $R$9 is the minimum number of inversions needed to transform $$\mathrm{inv}(L \rightarrow R)=\mathrm{inv}(L)+\mathrm{inv}(R),$$0 into an acyclic digraph; a sequence $$\mathrm{inv}(L \rightarrow R)=\mathrm{inv}(L)+\mathrm{inv}(R),$$1 achieving this is a decycling family (Alon et al., 2022). For oriented graphs $$\mathrm{inv}(L \rightarrow R)=\mathrm{inv}(L)+\mathrm{inv}(R),$$2 and $$\mathrm{inv}(L \rightarrow R)=\mathrm{inv}(L)+\mathrm{inv}(R),$$3, the dijoin $$\mathrm{inv}(L \rightarrow R)=\mathrm{inv}(L)+\mathrm{inv}(R),$$4 is the disjoint union of $$\mathrm{inv}(L \rightarrow R)=\mathrm{inv}(L)+\mathrm{inv}(R),$$5 and $$\mathrm{inv}(L \rightarrow R)=\mathrm{inv}(L)+\mathrm{inv}(R),$$6 together with an arc from every vertex of $$\mathrm{inv}(L \rightarrow R)=\mathrm{inv}(L)+\mathrm{inv}(R),$$7 to every vertex of $$\mathrm{inv}(L \rightarrow R)=\mathrm{inv}(L)+\mathrm{inv}(R),$$8 (Behague et al., 12 Sep 2025).

Setting	Central object	Conjectural statement
Packing of dijoins	Arc subsets meeting every dicut	Maximum number of disjoint dijoins $$\mathrm{inv}(L \rightarrow R)=\mathrm{inv}(L)+\mathrm{inv}(R),$$9 minimum dicut size $L \rightarrow R$0
Inversion number of dijoins	Join $L \rightarrow R$1 of oriented graphs	$L \rightarrow R$2

2. Woodall’s conjecture and its weighted analogue

Woodall’s conjecture states that if every directed cut of a digraph has at least $L \rightarrow R$3 edges, then there exist $L \rightarrow R$4 pairwise disjoint dijoins (Chudnovsky et al., 2014). The weighted analogue, due to Edmonds and Giles, asserts that in a weighted digraph the minimum weight of a dicut equals the maximum size of a packing of dijoins respecting the arc weights (Cornuéjols et al., 19 Jan 2025). The weighted statement is false in general, whereas the unweighted statement remains open (Cornuéjols et al., 2023).

A classical dual statement is the Lucchesi–Younger theorem: if every dijoin in a digraph has size at least $L \rightarrow R$5, then there are $L \rightarrow R$6 pairwise disjoint directed cuts (Chudnovsky et al., 2014). The asymmetry between the forward packing problem and its dual is central in the area: the dual theorem holds even in a capacitated version, but the capacitated extension of Woodall’s conjecture fails (Chudnovsky et al., 2014).

A standard obstruction comes from Schrijver’s counterexample. There exists a planar digraph $L \rightarrow R$7 and a subset $L \rightarrow R$8 such that every directed cut contains at least two edges in $L \rightarrow R$9, yet there do not exist two disjoint dijoins included in $L$0; in that example, the subdigraph formed by the edges in $L$1 consists of three disjoint paths (Chudnovsky et al., 2014). In all known counterexamples discussed there, the underlying undirected graph $L$2 is disconnected, which led to a connected-set variant of the conjecture (Chudnovsky et al., 2014).

3. Proven packing results and exact classes

A first general structural result is that if every dicut of $L$3 has size at least $L$4, then $L$5 can be partitioned into a dijoin and a $L$6-dijoin (Abdi et al., 2022). In the same work, for $L$7 and

$L$8

where each $L$9 is the integer in $R$0 equal to $R$1, the following are proved: if $R$2, then there is an equitable $R$3-weighted packing of dijoins of size $R$4; if $R$5, then there is a $R$6-weighted packing of dijoins of size $R$7; and if $R$8, $R$9, and $D=(V,A)$0, then $D=(V,A)$1 can be partitioned into three dijoins (Abdi et al., 2022).

A different line of work connects dijoin packing to nowhere-zero flows. If the underlying undirected graph of a digraph $D=(V,A)$2 with minimum dicut size $D=(V,A)$3 admits a nowhere-zero (circular) $D=(V,A)$4-flow, then $D=(V,A)$5 contains at least

$D=(V,A)$6

disjoint dijoins (Cornuéjols et al., 2023). Seymour’s existence of nowhere-zero $D=(V,A)$7-flows in $D=(V,A)$8-edge-connected graphs yields $D=(V,A)$9 disjoint dijoins, and these can be found in polynomial time; if the underlying undirected graph is $\delta^+(U)\subseteq A$0-edge-connected, the bound improves to $\delta^+(U)\subseteq A$1 disjoint dijoins (Cornuéjols et al., 2023).

Several exact positive results are known for restricted classes. For the connected-set variant at $\delta^+(U)\subseteq A$2, if $\delta^+(U)\subseteq A$3 and $\delta^+(U)\subseteq A$4 are compatible digraphs, $\delta^+(U)\subseteq A$5 is connected, and every directed cut of $\delta^+(U)\subseteq A$6 contains at least two edges from $\delta^+(U)\subseteq A$7, then $\delta^+(U)\subseteq A$8 can be partitioned into two dijoins whenever either $\delta^+(U)\subseteq A$9 is a caterpillar subdivision or $U\subsetneq V$0 is planar (Chudnovsky et al., 2014). In the weighted setting, the Edmonds–Giles conjecture is true if the underlying undirected graph is chordal, and there is a strongly polynomial time algorithm to construct such a packing (Cornuéjols et al., 19 Jan 2025). Since the unweighted case is a special case, this also gives the minimum-size-dicut $U\subsetneq V$1 maximum-number-of-disjoint-dijoins statement for chordal digraphs (Cornuéjols et al., 19 Jan 2025).

4. The inversion-number dijoin conjecture

For oriented graphs $U\subsetneq V$2 and $U\subsetneq V$3, the inversion-number dijoin conjecture of Bang-Jensen, da Silva, and Havet asserts

$U\subsetneq V$4

It is clear that

$U\subsetneq V$5

and earlier work had verified equality in special cases, including the cases where at least one component has inversion number zero and cases where both components are strongly connected with low inversion numbers (Alon et al., 2022).

The conjecture extends naturally to ordered $U\subsetneq V$6-joins. If $U\subsetneq V$7 denotes the generalized join obtained by adding all arcs from earlier factors to later ones, then one asks when

$U\subsetneq V$8

holds (Alon et al., 2022). This formulation has become central in subsequent classification results (Behague et al., 12 Sep 2025).

A later status summary isolates families where additivity is known: it holds when

$U\subsetneq V$9

where $\delta^-(U)=\emptyset$0 and $\delta^-(U)=\emptyset$1; counterexamples exist when either $\delta^-(U)=\emptyset$2 or $\delta^-(U)=\emptyset$3 is odd and at least $\delta^-(U)=\emptyset$4 and neither is zero; and the open cases are when both are even but not both $\delta^-(U)=\emptyset$5 (Wang et al., 2024).

5. Counterexamples and refined classifications for inversion number

The general conjecture is false. A decisive counterexample is a tournament $\delta^-(U)=\emptyset$6 such that

$\delta^-(U)=\emptyset$7

Since $\delta^-(U)=\emptyset$8, the conjectured value would have been $\delta^-(U)=\emptyset$9, so this shows strict inequality (Alon et al., 2022). The construction given there takes $R$00 to be a tournament on $R$01 vertices partitioned into three disjoint sets $R$02, $R$03, and $R$04, with the orientation chosen so that inverting $R$05, $R$06, and $R$07 yields an acyclic tournament (Alon et al., 2022).

At the same time, exact positive theorems delimit where additivity survives. If $R$08 and $R$09 satisfy $R$10, then

$R$11

and more generally, if $R$12 are oriented graphs with $R$13 for all $R$14, with equality for at most one $R$15, then

$R$16

The same work also gives a characterization of decycling families in $R$17-joins of inversion-$R$18 digraphs using orthonormal vectors over $R$19 (Alon et al., 2022).

Further refinements reveal a parity-sensitive pattern. If $R$20 is even and $R$21 is an oriented graph with $R$22, then

$R$23

equivalently, for all oriented graphs $R$24 with $R$25 and $R$26 even,

$R$27

That result disproves conjectures asserting that strict subadditivity should always occur once inversion numbers are large (Wang et al., 2024).

The classification is sharper when one factor has inversion number $R$28. If $R$29 and $R$30 is even, then

$R$31

If $R$32 and $R$33 is odd, then either

$R$34

or

$R$35

and the latter occurs precisely when

$R$36

The same paper proves an $R$37-join theorem: if all $R$38, then

$R$39

thereby proving a conjecture of Alon, Powierski, Savery, Scott, and Wilmer (Behague et al., 12 Sep 2025).

6. Techniques, computational complexity, and remaining frontier

The two conjectural frameworks have generated markedly different proof techniques. Approximate packing results use nowhere-zero and nowhere-zero circular $R$40-flows, together with reformulations of Woodall’s conjecture in terms of packing strongly connected orientations (Cornuéjols et al., 2023). The connected-set variant at $R$41 is reduced to an orientation problem on a tree $R$42 relative to a bias $R$43, with proofs in the caterpillar and planar cases using inductive orientation lemmas, planar duality, and the wedge theorem (Chudnovsky et al., 2014). Weighted packing results use reductions to weighted $R$44-bipartite digraphs and a matroidal framework involving the bimatchability matroid $R$45 and the major matroid $R$46 (Abdi et al., 2022). For chordal digraphs, the exact weighted theorem relies on simplicial vertices, perfect elimination orderings, and a weight-transfer argument (Cornuéjols et al., 19 Jan 2025).

In the inversion-number setting, linear algebra over $R$47 is central. One structural theorem characterizes decycling families in terms of orthonormal vectors over $R$48 (Alon et al., 2022). A later development introduces tournament minimum rank $R$49, defined as the minimal rank of a decycling matrix for a tournament $R$50, together with the relation

$R$51

and in the latter case $R$52 is even (Behague et al., 12 Sep 2025). This becomes the key algebraic tool in the classification of the $R$53 cases (Behague et al., 12 Sep 2025).

Complexity and extremal questions have also been clarified. For general oriented graphs, deciding whether $R$54 is NP-complete for all $R$55; for tournaments, deciding whether $R$56 is solvable in time $R$57, which is tight for all $R$58, and hence fixed-parameter tractable when parameterized by $R$59 (Alon et al., 2022). The maximum inversion number of an $R$60-vertex tournament is asymptotically $R$61 (Alon et al., 2022).

The present frontier is sharply delineated. In the packing setting, Woodall’s unweighted conjecture remains open, while the weighted Edmonds–Giles conjecture is false in general but true for chordal digraphs (Cornuéjols et al., 19 Jan 2025). In the inversion-number setting, counterexamples are known whenever one parameter is odd and at least $R$62, whereas the remaining open case is when both inversion numbers are even and at least $R$63 (Behague et al., 12 Sep 2025). A further conjecture is that tournament minimum rank is additive under dijoin,

$R$64

which, if true, would settle the even–even case via the relation between $R$65 and $R$66 (Behague et al., 12 Sep 2025).