Importance: High ✭✭✭
Author(s): Seymour, Paul D.
Recomm. for undergrads: yes
Posted by: nkorppi
on: October 9th, 2007
Conjecture   Any oriented graph has a vertex whose outdegree is at most its second outdegree.

By the $ n $th outdegree of $ v $, we mean the number of vertices for which the minimal outward-directed path from $ v $ to them is of length $ n $.

Chen, Shen, and Yuster [CSY] proved that in any oriented graph there is a vertex whose second outdegree is at least $ \gamma $ times its outdegree, where $ \gamma=0.657298... $ is the unique real root of $ 2x^3+x^2 -1=0 $.

This conjecture implies a special case of the \Oprefnum[Caccetta-Häggkvist Conjecture]{46385}.


[ASY] Chen, G.; Shen, J.; Yuster, R. Second neighborhood via first neighborhood in digraphs, Annals of Combinatorics, 7 (2003), 15--20.

[F] Fisher, David C. Squaring a tournament: a proof of Dean's conjecture. J. Graph Theory 23 (1996), no. 1, 43--48.

[KL] Kaneko, Yoshihiro; Locke, Stephen C. The minimum degree approach for Paul Seymour's distance 2 conjecture. Proceedings of the Thirty-second Southeastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 2001). Congr. Numer. 148 (2001), 201--206.

* indicates original appearance(s) of problem.


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