A tight Erdős-Pósa function for long cycles

Abstract A classic result of Erdős and Posa states that any graph either contains k vertex-disjoint cycles or can be made acyclic by deleting at most O ( k log ⁡ k ) vertices. Birmele, Bondy, and Reed (2007) raised the following more general question: given numbers l and k , what is the optimal function f ( l , k ) such that every graph G either contains k vertex-disjoint cycles of length at least l or contains a set X of f ( l , k ) vertices that meets all cycles of length at least l ? In this paper, we answer that question by proving that f ( l , k ) = Θ ( k l + k log ⁡ k ) . As a corollary, the tree-width of any graph G that does not contain k vertex-disjoint cycles of length at least l is of order O ( k l + k log ⁡ k ) . This is also optimal up to constant factors and answers another question of Birmele, Bondy, and Reed (2007).

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