Geodesic
On directed triangles in digraphs
The electronic journal of combinatorics, Tome 14 (2007)
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Using a recent result of Chudnovsky, Seymour, and Sullivan, we slightly improve two bounds related to the Caccetta-Haggkvist Conjecture. Namely, we show that if $\alpha\geq 0.35312$, then each $n$-vertex digraph $D$ with minimum outdegree at least $\alpha n$ has a directed $3$-cycle. If $\beta\geq 0.34564$, then every $n$-vertex digraph $D$ in which the outdegree and the indegree of each vertex is at least $\beta n$ has a directed $3$-cycle.
DOI : 10.37236/1020
Classification : 05C20, 05C35
Mots-clés : digraph, minimum outdegree, derected cycle, indegree, outdegree 
@article{10_37236_1020,
     author = {Peter Hamburger and Penny Haxell and Alexandr Kostochka},
     title = {On directed triangles in digraphs},
     journal = {The electronic journal of combinatorics},
     year = {2007},
     volume = {14},
     doi = {10.37236/1020},
     zbl = {1157.05311},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1020/}
}
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Peter Hamburger; Penny Haxell; Alexandr Kostochka. On directed triangles in digraphs. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/1020

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