Zero-Sum Cycles in Regular Digraphs

Abstract: Let $Γ$ be a finite group of order $k\ge2$, and label the edges of a simple loopless $d$-regular digraph $D$ by elements of $Γ$. A directed cycle is zero-sum if the ordered product of its labels is the identity of $Γ$. We prove that a zero-sum cycle exists whenever $d\ge e<sup>3(k-1)$. We also prove that every labelled $d$-regular digraph contains $Ω(d/k)$ pairwise vertex-disjoint zero-sum cycles. When $d\ge50k$, it contains $Ω(d<sup>2/k)$ pairwise edge-disjoint zero-sum cycles. All three results are asymptotically optimal. The existence and packing results extend to Eulerian digraphs whose minimum and maximum common degrees $δ$ and $Δ$ satisfy $δ<sup>3/Δ<sup>2=Ω(k)$. The techniques extend a determinant--permanent argument of Friedland for even directed cycles.

• The paper proves that every d-regular digraph labelled by a finite group of order k contains a zero-sum directed cycle when d ≥ e³(k−1), with the linear dependence on k optimal up to constants.
• It develops a block-determinant method using centered permutation matrices to encode group products as traces, extending determinant–permanent techniques beyond parity-based cycle results.
• It establishes packings of ⌊d/(25k)⌋ vertex-disjoint and at least d²/(100k) edge-disjoint zero-sum cycles, while extending analogous results to Eulerian digraphs when δ³/Δ² = Ω(k).

Overview

The paper proves that sufficiently dense regular digraphs force zero-sum directed cycles under arbitrary finite-group edge labellings, and gives asymptotically optimal packing bounds for such cycles. For a finite group $\Gamma$ of order $k\ge 2$ labelling the edges of a simple loopless digraph $D$, a directed cycle is zero-sum if the ordered product of its edge labels is the identity; the condition is independent of the chosen initial vertex because changing it conjugates the label product. The main theorems are: every $d$-regular $\Gamma$-labelled digraph with $d\ge e^3(k-1)$ contains a zero-sum cycle; it contains $\lfloor d/(25k)\rfloor$ pairwise vertex-disjoint zero-sum cycles; and, for $d\ge 50k$, it contains at least $d^2/(100k)$ pairwise edge-disjoint zero-sum cycles. All three bounds are sharp up to constants, and the existence and packing results extend to Eulerian digraphs under the degree-ratio condition $\delta^3/\Delta^2=\Omega(k)$, where $k\ge 2$0 and $k\ge 2$1 are the minimum and maximum common degrees (2608.14515).

The method is a genuine extension of Friedland's determinant–permanent argument for even directed cycles, which previously could not handle groups of order $k\ge 2$2: signs distinguish even from odd cycles, but a group-labelled cycle requires distinguishing the identity product from every nonidentity product, and determinant signs provide no such distinction. The paper resolves this by replacing scalar entries with $k\ge 2$3 group-action matrices, so that the trace of the product of blocks along a cycle encodes the cycle's label product.

The determinant identity

The central technical device is a squarefree version of a block-determinant identity in the spirit of MacMahon's master theorem, Chu's determinant–permanent correspondence, and the trace identities of Kassel and Lévy. Working in the squarefree algebra $k\ge 2$4, define the cycle-collection polynomial $k\ge 2$5 over the simple directed cycles of $k\ge 2$6; its squarefree coefficients count directed cycle covers, so $k\ge 2$7.

Given a $k\ge 2$8 matrix $k\ge 2$9 on every edge, the paper forms the block matrix $D$0 and the determinant polynomial $D$1, where $D$2. The identity states that in $D$3,

$D$4

where $D$5 is the ordered product of edge blocks along $D$6 and the trace is well defined by cyclicity. The proof extracts surviving permutations in the determinant expansion: since looplessness forces $D$7, only permutations moving at most one index pair per vertex survive the squarefree reduction, and the sign computation $D$8 per cycle converts the determinant into a product of trace terms. Crucially, $D$9 as an ordinary polynomial, which is what makes coefficient extraction via roots-of-unity averaging and Hadamard's inequality possible.

The group enters through the centered permutation matrices $d$0, where $d$1 is the permutation matrix of $d$2 on $d$3. These satisfy $d$4 and $d$5 if $d$6 and $d$7 otherwise. Assigning $d$8 therefore gives $d$9 exactly for zero-sum cycles and $\Gamma$0 otherwise. If $\Gamma$1 has no zero-sum cycle, every cycle factor becomes $\Gamma$2, so $\Gamma$3 in $\Gamma$4 and every $\Gamma$5 is a squarefree coefficient of $\Gamma$6.

The existence threshold $\Gamma$7

The regular existence proof is a direct comparison. If $\Gamma$8 is $\Gamma$9-regular on $d\ge e^3(k-1)$0 vertices, the van der Waerden theorem applied to $d\ge e^3(k-1)$1 gives $d\ge e^3(k-1)$2. Conversely, since each row of $d\ge e^3(k-1)$3 has norm $d\ge e^3(k-1)$4, a roots-of-unity averaging argument (in the style of Lossers' solution to a Goldstein–Graham problem) together with Hadamard's inequality yields $d\ge e^3(k-1)$5 with $d\ge e^3(k-1)$6 and $d\ge e^3(k-1)$7, i.e. $d\ge e^3(k-1)$8. The two bounds are incompatible when $d\ge e^3(k-1)$9, so a zero-sum cycle must exist.

The linear dependence on $\lfloor d/(25k)\rfloor$0 is necessary: the paper constructs, for every $\lfloor d/(25k)\rfloor$1 and $\lfloor d/(25k)\rfloor$2, a strongly connected $\lfloor d/(25k)\rfloor$3-regular circulant digraph on $\lfloor d/(25k)\rfloor$4 vertices with a $\lfloor d/(25k)\rfloor$5-labelling (labels record whether an edge wraps a cyclic order) in which every directed cycle has nonzero label sum. This shows no bound depending on $\lfloor d/(25k)\rfloor$6 can drop below order $\lfloor d/(25k)\rfloor$7, and it also shows the degree $\lfloor d/(25k)\rfloor$8 in the paper's conjecture for strongly connected digraphs is best possible uniformly over groups of order $\lfloor d/(25k)\rfloor$9.

Compared with the Alon–Linial bound, which via the Lovász local lemma requires $d\ge 50k$0, the new theorem allows $d\ge 50k$1 of order $d\ge 50k$2 when $d\ge 50k$3 is bounded—removing a logarithmic loss. The comparison is honest about its limits: for degree ratios $d\ge 50k$4 above roughly $d\ge 50k$5, the Alon–Linial bound remains stronger and also applies without the Eulerian assumption.

Packing bounds

For vertex-disjoint cycles, the paper considers the generating polynomial $d\ge 50k$6 over directed cycle covers, where $d\ge 50k$7 counts zero-sum cycles. Since $d\ge 50k$8, the van der Waerden bound gives $d\ge 50k$9. If every cover had fewer than $d^2/(100k)$0 zero-sum cycles, then $d^2/(100k)$1, and a finite-difference (interpolation) identity expresses $d^2/(100k)$2 in terms of the values $d^2/(100k)$3 for $d^2/(100k)$4. Each $d^2/(100k)$5 is realized as the coefficient $d^2/(100k)$6 of a determinant built from block-diagonal matrices $d^2/(100k)$7, which satisfy $d^2/(100k)$8 for zero-sum cycles and $d^2/(100k)$9 otherwise. Hadamard bounds each value by $\delta^3/\Delta^2=\Omega(k)$0; combining with binomial weights gives $\delta^3/\Delta^2=\Omega(k)$1, a contradiction. The same technique applied to a weighted polynomial $\delta^3/\Delta^2=\Omega(k)$2 over partial cycle collections yields the Eulerian analogue $\delta^3/\Delta^2=\Omega(k)$3 vertex-disjoint zero-sum cycles.

Edge-disjoint packings follow by iteration: deleting a directed cycle cover from a $\delta^3/\Delta^2=\Omega(k)$4-regular digraph leaves a $\delta^3/\Delta^2=\Omega(k)$5-regular digraph, so summing the vertex-disjoint guarantee over $\delta^3/\Delta^2=\Omega(k)$6 gives at least $\delta^3/\Delta^2=\Omega(k)$7 pairwise edge-disjoint zero-sum cycles; the Eulerian version gives $\delta^3/\Delta^2=\Omega(k)$8 under $\delta^3/\Delta^2=\Omega(k)$9.

These orders are sharp. The balanced blow-up of a directed cycle on $k\ge 2$00 classes of size $k\ge 2$01, labelled constantly by $k\ge 2$02 with $k\ge 2$03, has every zero-sum cycle of length at least $k\ge 2$04, hence at most $k\ge 2$05 vertex-disjoint and $k\ge 2$06 edge-disjoint such cycles; if $k\ge 2$07 it has none. The paper's constants (25, 100, $k\ge 2$08, $k\ge 2$09) are therefore not optimized, but the dependence on $k\ge 2$10 and $k\ge 2$11 cannot be improved. Relative to the previously best packing bounds derivable from Alon's disjoint-cycle theorem combined with Alon–Linial, which required $k\ge 2$12 and gave $k\ge 2$13 and $k\ge 2$14, the new results remove the factor $k\ge 2$15 from both the threshold and the guarantees.

The Eulerian extension

For irregular Eulerian digraphs, $k\ge 2$16 itself may vanish, so the argument shifts to the diagonally weighted permanent of $k\ge 2$17. Two estimates are compared. For the upper bound, a weighted roots-of-unity extraction expresses $k\ge 2$18 as an expectation of determinant evaluations, and Hadamard's inequality gives, for every $k\ge 2$19,

$k\ge 2$20

For the lower bound, the paper uses Gurvits's Bethe permanent inequality with an explicit doubly stochastic matrix $k\ge 2$21 built from the degree profile: $k\ge 2$22 on edges and $k\ge 2$23 on the diagonal. With the choices $k\ge 2$24 and $k\ge 2$25 where $k\ge 2$26, the per-vertex ratio of the Bethe lower bound to the Hadamard upper bound exceeds

$k\ge 2$27

whenever $k\ge 2$28, a contradiction. The result reduces to the regular theorem at $k\ge 2$29.

A corollary of independent interest: every $k\ge 2$30-regular digraph with $k\ge 2$31 contains pairwise edge-disjoint cycles $k\ge 2$32 with $k\ge 2$33 dividing $k\ge 2$34 for each $k\ge 2$35, obtained by peeling cycles of decreasing modulus while tracking the Eulerian degree-ratio condition through the deletions.

Limitations and open problems

The paper is explicit about the boundaries of its results. The existence and packing theorems require regularity or, more generally, the Eulerian condition with bounded $k\ge 2$36; minimum indegree and outdegree alone do not force even an unlabelled even cycle, as the constructions of Thomassen and of Gutin–Sudakov–Yeo show, so no analogue can hold in that generality. The constants in all bounds are non-sharp, and the degree threshold $k\ge 2$37 is presumably far from optimal. Two questions are left open. First, the paper conjectures that every strongly connected digraph with minimum indegree and outdegree at least $k\ge 2$38 contains a zero-sum cycle under every $k\ge 2$39-labelling—sharp by the circulant obstruction, and a strengthening of Diwan's $k\ge 2$40-version; since weakly connected Eulerian digraphs are strongly connected, this would replace the degree-ratio hypothesis by the sharp condition $k\ge 2$41. Second, it asks whether the exponent $k\ge 2$42 in the Eulerian condition $k\ge 2$43 can be reduced below $k\ge 2$44 (the paper proves $k\ge 2$45; the conjectured strongly connected threshold corresponds to $k\ge 2$46). It also notes that in the complete-digraph setting, elementary abelian groups admit bounds of order $k\ge 2$47 rather than $k\ge 2$48, and asks whether analogous exponent-dependent bounds exist in the regular setting.

Conclusion

The paper settles the Alon–Linial question for regular digraphs with an asymptotically optimal answer: $k\ge 2$49 forces a zero-sum cycle under every finite-group labelling, with matching optimal-order packings of $k\ge 2$50 vertex-disjoint and $k\ge 2$51 edge-disjoint zero-sum cycles, and extends everything to Eulerian digraphs under the condition $k\ge 2$52. Technically, it shows that the determinant–permanent method, previously confined to parity through signs, extends to arbitrary finite groups by encoding label products as traces of centered permutation matrices, combined with Bethe permanent lower bounds and finite-difference interpolation for the packing and irregular cases. The main structural open question is whether the Eulerian degree-ratio hypothesis can be weakened toward the conjecturally sharp strongly connected threshold $k\ge 2$53 (2608.14515).