Degree condition for maximum-length cycles in bipartite digraphs

Let DD be a bipartite digraph with colour classes XX and YY such that X=ab=Y|X|=a\leq b=|Y|. For vertices uu and vv in opposite colour classes with uvA(D)uv\notin A(D), consider the degree condition

d+(u)+d(v)>a+b+22.d^+(u)+d^-(v)>\frac{a+b+2}{2}.

Maximum-cycle degree conjecture. If this inequality holds whenever uu and vv lie in opposite colour classes and uvA(D)uv\notin A(D), then DD contains an oriented cycle of length 2a2a.

This is an Ore-type sufficient condition for a bipartite digraph to contain a cycle meeting every vertex of its smaller colour class. The source states that the conjecture is proved in the balanced case a=ba=b, while the general case remains open.

Sources & referencesView supporting material

Primary source

Janusz Adamus and Lech Adamus, “A degree condition for cycles of maximum length in bipartite digraphs”, arXiv:1101.4973 (2012).

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