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- W2803601511 abstract "Judicious partitioning problems on graphs ask for partitions that bound several quantities simultaneously, which have received a lot of attentions lately. Scott asked the following natural question: What is the maximum constant $c_d$ such that every directed graph $D$ with $m$ arcs and minimum outdegree $d$ admits a bipartition $V(D)= V_1cup V_2$ satisfying $min{e(V_1, V_2), e(V_2, V_1)}ge c_d m$? Here, for $i=1,2$, $e(V_{i},V_{3-i})$ denotes the number of arcs in $D$ from $V_{i}$ to $V_{3-i}$. Lee, Loh, and Sudakov %[Judicious partitions of directed graphs, Random Struct. Alg. 48 %(2016) 147--170] conjectured that every directed graph $D$ with $m$ arcs and minimum outdegree at least $dge 2$ admits a bipartition $V(D)=V_1cup V_2$ such that [ min{e(V_1,V_2),e(V_2,V_1)}geq Big(frac{d-1}{2(2d-1)}+ o(1)Big)m. ] %While it is not known whether or not the minimum outdegree condition %alone is sufficient, w We show that this conjecture holds under the additional natural condition that the minimum indegree is also at least $d$." @default.
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- W2803601511 date "2018-05-14" @default.
- W2803601511 modified "2023-09-27" @default.
- W2803601511 title "A bound on judicious bipartitions of directed graphs" @default.
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- W2803601511 doi "https://doi.org/10.48550/arxiv.1805.05506" @default.
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