Gishboliner–Steiner–Szabó conjecture on subdivisions of the bidirected triangle

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Let FF be a digraph, and let maderχ⃗(F)\mathrm{mader}_{\vec{\chi}}(F) be the least integer kk such that every digraph DD with dichromatic number χ⃗(D)≥k\vec{\chi}(D)\geq k contains a subdivision of FF. Write K3↔\overleftrightarrow{K_3} for the digraph obtained from the complete graph K3K_3 by replacing each edge with both possible directed arcs. Gishboliner–Steiner–Szabó conjecture.

maderχ⃗(K3↔)=4.\mathrm{mader}_{\vec{\chi}}(\overleftrightarrow{K_3})=4.

The exact value of maderχ⃗(F)\mathrm{mader}_{\vec{\chi}}(F) is known for only a few digraphs, and K3↔\overleftrightarrow{K_3} is the smallest digraph for which the value was unknown in the source. The conjecture is presented as an open problem.

References

Primary source

Lucas Picasarri-Arrieta and Clément Rambaud, “Subdivisions in dicritical digraphs with large order or digirth”, arXiv:2401.05938 (2024).

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