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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2311.04337 |
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| _version_ | 1866908373663875072 |
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| author | Cornuéjols, Gérard Liu, Siyue Ravi, R. |
| author_facet | Cornuéjols, Gérard Liu, Siyue Ravi, R. |
| contents | In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects each dicut. Woodall conjectured in 1976 that in every digraph, the minimum size of a dicut equals to the maximum number of disjoint dijoins. However, prior to our work, it was not even known whether at least $3$ disjoint dijoins exist in an arbitrary digraph whose minimum dicut size is sufficiently large. By building connections with nowhere-zero (circular) $k$-flows, we prove that every digraph with minimum dicut size $τ$ contains $\left\lfloor\fracτ{k}\right\rfloor$ disjoint dijoins if the underlying undirected graph admits a nowhere-zero (circular) $k$-flow. The existence of nowhere-zero $6$-flows in $2$-edge-connected graphs (Seymour 1981) directly leads to the existence of $\left\lfloor\fracτ{6}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $τ$, which can be found in polynomial time as well. The existence of nowhere-zero circular $\frac{2p+1}{p}$-flows in $6p$-edge-connected graphs (Lovász et al. 2013) directly leads to the existence of $\left\lfloor\frac{τp}{2p+1}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $τ$ whose underlying undirected graph is $6p$-edge-connected. We also discuss reformulations of Woodall's conjecture into packing strongly connected orientations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_04337 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Approximately Packing Dijoins via Nowhere-Zero Flows Cornuéjols, Gérard Liu, Siyue Ravi, R. Combinatorics In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects each dicut. Woodall conjectured in 1976 that in every digraph, the minimum size of a dicut equals to the maximum number of disjoint dijoins. However, prior to our work, it was not even known whether at least $3$ disjoint dijoins exist in an arbitrary digraph whose minimum dicut size is sufficiently large. By building connections with nowhere-zero (circular) $k$-flows, we prove that every digraph with minimum dicut size $τ$ contains $\left\lfloor\fracτ{k}\right\rfloor$ disjoint dijoins if the underlying undirected graph admits a nowhere-zero (circular) $k$-flow. The existence of nowhere-zero $6$-flows in $2$-edge-connected graphs (Seymour 1981) directly leads to the existence of $\left\lfloor\fracτ{6}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $τ$, which can be found in polynomial time as well. The existence of nowhere-zero circular $\frac{2p+1}{p}$-flows in $6p$-edge-connected graphs (Lovász et al. 2013) directly leads to the existence of $\left\lfloor\frac{τp}{2p+1}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $τ$ whose underlying undirected graph is $6p$-edge-connected. We also discuss reformulations of Woodall's conjecture into packing strongly connected orientations. |
| title | Approximately Packing Dijoins via Nowhere-Zero Flows |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2311.04337 |