TY - JOUR

T1 - Large feedback arc sets, high minimum degree subgraphs, and long cycles in eulerian digraphs

AU - Huang, Hao

AU - Ma, Jie

AU - Shapira, Asaf

AU - Sudakov, Benny

AU - Yuster, Raphael

PY - 2013/11

Y1 - 2013/11

N2 - A minimum feedback arc set of a directed graph G is a smallest set of arcs whose removal makes G acyclic. Its cardinality is denoted by β(G). We show that a simple Eulerian digraph with n vertices and m arcs has β(G) ≥ m 2/2n 2+m/2n, and this bound is optimal for infinitely many m, n. Using this result we prove that a simple Eulerian digraph contains a cycle of length at most 6n 2/m, and has an Eulerian subgraph with minimum degree at least m 2/24n 3. Both estimates are tight up to a constant factor. Finally, motivated by a conjecture of Bollobás and Scott, we also show how to find long cycles in Eulerian digraphs.

AB - A minimum feedback arc set of a directed graph G is a smallest set of arcs whose removal makes G acyclic. Its cardinality is denoted by β(G). We show that a simple Eulerian digraph with n vertices and m arcs has β(G) ≥ m 2/2n 2+m/2n, and this bound is optimal for infinitely many m, n. Using this result we prove that a simple Eulerian digraph contains a cycle of length at most 6n 2/m, and has an Eulerian subgraph with minimum degree at least m 2/24n 3. Both estimates are tight up to a constant factor. Finally, motivated by a conjecture of Bollobás and Scott, we also show how to find long cycles in Eulerian digraphs.

KW - 2010 Mathematics subject classification: Primary 05C20

KW - Secondary 05C38

UR - http://www.scopus.com/inward/record.url?scp=84893073626&partnerID=8YFLogxK

U2 - 10.1017/S0963548313000394

DO - 10.1017/S0963548313000394

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AN - SCOPUS:84893073626

SN - 0963-5483

VL - 22

SP - 859

EP - 873

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 6

ER -