Jungho Ahn (안정호), A coarse Erdős-Pósa theorem for constrained cycles

An induced packing of cycles in a graph is a set of vertex-disjoint cycles such that the graph has no edge between distinct cycles of the set. The classic Erdős-Pósa theorem shows that for every positive integer $k$, every graph contains $k$ vertex-disjoint cycles or a set of $O(k\log k)$ vertices which intersects every cycle of $G$.

We generalise this classic Erdős-Pósa theorem to induced packings of cycles of length at least $\ell$ for any integer $\ell$. We show that there exists a function $f(k,\ell )=O(\ell k\log k)$ such that for all positive integers $k$ and $\ell$ with $\ell \ge 3$, every graph $G$ contains an induced packing of $k$ cycles of length at least $\ell$ or a set $X$ of at most $f(k,\ell )$ vertices such that the closed neighbourhood of $X$ intersects every cycle of $G$.

Furthermore, we extend the result to long cycles containing prescribed vertices. For a graph $G$ and a set $S\subseteq V(G)$, an $S$-cycle in $G$ is a cycle containing a vertex in $S$. We show that for all positive integers $k$ and $\ell$ with $\ell \ge 3$, every graph $G$, and every set $S\subseteq V(G)$, $G$ contains an induced packing of $k$ $S$-cycles of length at least $\ell$ or a set $X$ of at most $\ell k^{O(1)}$ vertices such that the closed neighbourhood of $X$ intersects every cycle of $G$.

Our proofs are constructive and yield polynomial-time algorithms, for fixed $\ell$, finding either the induced packing of the constrained cycles or the set $X$.

This is based on joint works with Pascal Gollin, Tony Huynh, and O-joung Kwon.