Strongly connected zero-sum cycle threshold

Prove that every strongly connected digraph with minimum indegree and minimum outdegree at least k+1 contains a zero-sum directed cycle under every edge-labelling by a finite group of order k.

Background

The paper establishes zero-sum cycle existence for d-regular digraphs when d is at least e3(k-1), and for Eulerian digraphs under the degree-ratio condition δ3/Δ2 ≥ e3(k-1). However, minimum indegree and minimum outdegree alone do not force a zero-sum cycle without additional structural assumptions, even when both parameters are arbitrarily large.

The proposed conjecture asks whether strong connectivity suffices with the sharp threshold k+1 for every finite group of order k. The threshold cannot be lowered uniformly over all such groups, because the paper constructs strongly connected k-regular digraphs with a Z_k-labelling and no zero-sum directed cycle. The conjecture would also imply the corresponding sharp result for Eulerian digraphs, since every weakly connected Eulerian digraph is strongly connected.

References

In view of our results for arbitrary finite groups, we propose the following stronger version with the conjecturally sharp threshold, which agrees with the complete case.

Let $\Gamma$ be a finite group of order $k\ge2$. Every strongly connected digraph with minimum indegree and minimum outdegree at least $k+1$ contains a zero-sum directed cycle under every $\Gamma$-labelling of its edges.

— Zero-Sum Cycles in Regular Digraphs  (2608.14515 - Sivashankar, 14 Aug 2026) in Conjecture 7.1, Section 7, “Discussion and open problems”

A weaker intermediate question is whether the exponent $2$ in the degree-ratio condition of Theorem~\ref{thm:eulerian} can be reduced.

Do there exist absolute constants $0\le\alpha<2$ and $C>0$ such that for every finite group $\Gamma$ of order $k\ge2$, every Eulerian digraph with minimum and maximum common degrees $\delta$ and $\Delta>0$ contains a zero-sum directed cycle under every $\Gamma$-labelling whenever

\frac{\delta{1+\alpha}}{\Delta\alpha}\ge Ck?

— Zero-Sum Cycles in Regular Digraphs  (2608.14515 - Sivashankar, 14 Aug 2026) in Question 7.2, Section 7, “Discussion and open problems”