Lichiardopol’s Conjecture on Disjoint Cycles in Tournaments

Abstract

In 2010, Lichiardopol conjectured for $q \geqslant 3$ and $k \geqslant 1$ that any tournament with minimum out-degree at least $(q-1)k-1$ contains $k$ disjoint cycles of length $q$. Previously the conjecture was known to hold for $q\leqslant 4$. We prove that it holds for $q \geqslant 5$, thereby completing the proof of the conjecture.

Vertex‐disjoint cycles of the same length in tournaments

“…This asymptotically answers a question “whether there is a positive constant $$\epsilon $$ such that the minimum outdegree $$(q\unicode{x02215}2+\epsilon )k$$ suffices to guarantee $$k$$ disjoint $$q$$‐cycles when $$q$$ is sufficiently large” in [15]. Indeed, in the same paper, Ma, West, and Yan gave a lower bound of $$(q-1)k-1$$ on the minimum outdegree for $$k$$ disjoint $$q$$‐cycles in tournaments, which confirmed a conjecture due to Lichiardopol [11] for $$q\ge 5$$.…”

“…Let $$T$$ be a tournament of order $$n$$ with $${\delta }^{+}(T)\ge \alpha k$$. We assume that $$\alpha \lt q-1$$, otherwise Theorem 2 is directly proved in [15, 17]. First we take as many disjoint $$q$$‐cycles in $$T$$ as possible.…”