Degree conditions forcing directed cycles

Abstract

Caccetta-Häggkvist conjecture is a longstanding open problem on degree conditions that force an oriented graph to contain a directed cycle of a bounded length. Motivated by this conjecture, Kelly, Kühn, and Osthus initiated a study of degree conditions forcing the containment of a directed cycle of a given length. In particular, they found the optimal minimum semidegree, that is, the smaller of the minimum indegree and the minimum outdegree, which forces a large oriented graph to contain a directed cycle of a given length not divisible by 3, and conjectured the optimal minimum semidegree for all the other cycles except the directed triangle. In this paper, we establish the best possible minimum semidegree that forces a large oriented graph to contain a directed cycle of a given length divisible by 3 yet not equal to 3, hence fully resolve the conjecture by Kelly, Kühn, and Osthus. We also find an asymptotically optimal semidegree threshold of any cycle with a given orientation of its edges with the sole exception of a directed triangle.

Figures (12)

• Figure 1: The $2$-shortcut, the $3$-shortcut, the $4$-shortcut and a general $s$-shortcut.
• Figure 2: Maneuvers used when $k$ is odd. In the first one, adding the middle vertex and incident edges, which creates cycles of length $(k+1)/2$ and $(k-1)/2$, allows to remove a vertex in the leftmost blob and in the rightmost blob. In the remaining two maneuvers, adding edges to an arbitrary vertex in each of the two bottommost blobs allows to remove a vertex in the topmost blob.
• Figure 3: Maneuvers used when $k$ is even. In the first one, adding the middle vertex and incident edges allows to remove one vertex in each of the two bottommost blobs. In the other one, adding the middle two vertices and incident edges allows to remove a vertex in each blob except the bottom right one.
• Figure 4: Setting used in the proof that there is no path of length smaller than $5$ from the sink of a transitive triangle to its source.
• Figure 5: Partitioning of the graph vertices into the sets $A$, $B$, $C$, and $D$.

Theorems & Definitions (27)

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• proof : Proof of Theorem \ref{['thm:kbig_walk']}
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• proof : Proof of \ref{['thm:k4walk']}