Abstract

In this note, it is proved that if α≥0.24817, then any digraph on n vertices with minimum outdegree at least αn contains a directed cycle of length at most 5.

Highlights

  • In 1978, Caccetta and Häggkvist [1] made the following conjecture: Conjecture 1.1 Any digraph on n vertices with minimum outdegree at least r contains a directed cycle of length at most n r

  • By refining the combinatorial techniques in [6,7,11], we prove the following result

  • Substituting (3), (4) and (5) into this inequalities yields n r N w \ N v

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Summary

Introduction

It is proved that if 0.24817 , any digraph on n vertices with minimum outdegree at least n contains a directed cycle of length at most 5. In 1978, Caccetta and Häggkvist [1] made the following conjecture: Conjecture 1.1 Any digraph on n vertices with minimum outdegree at least r contains a directed cycle of length at most n r . One may seek as small a constant as possible such that any digraph on n vertices with minimum outdegree at least n contains a directed triangle.

Results
Conclusion

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