Li's directed C4C_4 conjecture for oriented bipartite graphs

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Let DD be an oriented bipartite graph with bipartition (A,B)(A,B), and let d+(x)d^+(x) denote the out-degree of a vertex xx. A directed C4C_4 is a directed cycle of length four in DD. Li's conjecture. If

d+(u)>∣B∣/3d^{+}(u)>|B|/3

for each u∈Au\in A and

d+(v)>∣A∣/3d^{+}(v)>|A|/3

for each v∈Bv\in B, then DD has a directed C4C_4. H. Li proposed this conjecture and proved it for balanced oriented bipartite graphs; the general statement is not resolved by the supplied text.

References

Primary source

Bo Ning and Jun Ge, “Rainbow C_4's and Directed C_4's: the Bipartite Case Study”, arXiv:1301.5697 (2013).

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