A note on complete subdivisions in digraphs of large outdegree

Authors:
Daniela Kühn;Deryk Osthus;Andrew Young
Affiliations:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK;School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK;School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
Venue:
Journal of Graph Theory
Year:
2008

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Abstract

Mader conjectured that for all ℓ there is aninteger δ+(ℓ) such that every digraphof minimum outdegree at leastδ+(ℓ)contains a subdivision of atransitive tournament of order ℓ. In this note, weobserve that if the minimum outdegree of a digraph is sufficientlylarge compared to its order then one can even guarantee asubdivision of a large complete digraph. More precisely, let $\vecG$ be a digraph of order n whose minimum outdegree is atleast d. Then $\vec G$ contains a subdivision of a completedigraph of order[d2/8n3/2]. © 2007Wiley Periodicals, Inc. J Graph Theory 57: 16, 2008.