Degree conditions forcing directed cycles

TL;DR

This paper resolves a key conjecture by establishing the exact minimum semidegree conditions needed to force large oriented graphs to contain directed cycles of specified lengths divisible by 3, excluding the triangle.

Contribution

It determines the optimal semidegree threshold for cycles of lengths divisible by 3, completing the previous conjecture and extending understanding of degree conditions for directed cycles.

Findings

Established the best possible semidegree for cycles divisible by 3, excluding 3.

Provided asymptotic thresholds for cycles with specific orientations.

Resolved the Kelly-K"uhn-Osthus conjecture for all cycle lengths except the directed triangle.

Abstract

Caccetta-H\"{a}ggkvist 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, K\"uhn, 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, K\"{u}hn, 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.