The mixed Nine Dragon Tree conjecture for spanning mixed arborescences

Let H=(V,E,A)H=(V,E,A) be a mixed graph, let D(V)\mathcal{D}(V) denote the relevant family of subpartitions, and let dA−(X)d_A^-(X) count arcs entering XX. Let k≥0k\geq0 and d≥1d\geq1 be integers. Mixed Nine Dragon Tree conjecture. If, for every P∈D(V)\mathcal{P}\in\mathcal{D}(V),

E(P)+∑X∈PdA−(X)∣P∣−1>k+d−1d,\frac{E(\mathcal{P})+\sum_{X\in\mathcal{P}}d_A^-(X)}{|\mathcal{P}|-1}>k+\frac{d-1}{d},

then HH contains edge- and arc-disjoint kk spanning mixed arborescences and another mixed branching FF satisfying

∣E(F)∣+∣A(F)∣>d−1d(∣V∣−1).|E(F)|+|A(F)|>\frac{d-1}{d}(|V|-1).

If FF is not a spanning arborescence, then FF has a component with at least dd edges and arcs. Moreover, the bound is sharp. This would extend the spanning-arborescence packing results from graphs and digraphs to mixed graphs; no resolution is supplied in the text.

References

Primary source

Hui Gao, “Packing spanning arborescences with extra large one”, arXiv:2511.18952 (2025).

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