2 research outputs found
A bound on judicious bipartitions of directed graphs
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 such that every directed graph with arcs and minimum
outdegree admits a bipartition satisfying ? Here, for , denotes
the number of arcs in from to . Lee, Loh, and Sudakov
%[Judicious partitions of directed graphs, Random Struct. Alg. 48 %(2016)
147--170] conjectured that every directed graph with arcs and minimum
outdegree at least admits a bipartition such that
%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 .Comment: 16 page
Partitioning digraphs with outdegree at least 4
Scott asked the question of determining such that if is a digraph
with arcs and minimum outdegree then has a partition such that , where
(respectively, ) is the number of arcs from to
(respectively, from to ). Lee, Loh, and Sudakov showed that
and , and conjectured that for . In this paper, we show
and prove some partial results for .Comment: 18 page