Partitioning digraphs with outdegree at least 4

Abstract

Scott asked the question of determining (Formula presented.) such that if (Formula presented.) is a digraph with (Formula presented.) arcs and minimum outdegree (Formula presented.) then (Formula presented.) has a partition (Formula presented.) such that (Formula presented.), where (Formula presented.) (respectively, (Formula presented.)) is the number of arcs from (Formula presented.) to (Formula presented.) (respectively, from (Formula presented.) to (Formula presented.)). Lee, Loh, and Sudakov showed that (Formula presented.) and (Formula presented.), and conjectured that (Formula presented.) for (Formula presented.). In this paper, we show (Formula presented.) and prove some partial results for (Formula presented.).