Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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(klog⁡k)O(klog⁡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)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)f(l,k) vertices that meets all cycles of length at least l  ? ,  ansr that question by proving that f(l,k)=Θ(kl+klog⁡k)f(l,k)=Θ(kl+klog⁡k).#R##N##R##N#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(kl+klog⁡k)O(kl+klog⁡k). This is also optimal up to constant factors and ansrs another question of Birmele, Bondy, and Reed (2007).
['Frank Mousset', 'Andreas Noever', 'Nemanja Škorić', 'Felix Weissenberger']
I am writing a research paper on a idea A tight Erdős-Pósa function for long cycles
[{"from": "human", "value": "I am writing a research paper on a idea A tight Erd\u0151s-P\u00f3sa function for long cycles"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A classic result of Erd\u0151s and Posa states that any graph either contains k \u00a0 vertex-disjoint cycles or can be made acyclic by deleting at most O(klog\u2061k)O(klog\u2061k) vertices. Birmele, Bondy, and Reed (2007) raised the following more general question: given numbers l and k \u00a0, what is the optimal function f(l,k)f(l,k) such that every graph G either contains k vertex-disjoint cycles of length at least l or contains a set X \u00a0 of f(l,k)f(l,k) vertices that meets all cycles of length at least l \u00a0? ,  ansr that question by proving that f(l,k)=\u0398(kl+klog\u2061k)f(l,k)=\u0398(kl+klog\u2061k).#R##N##R##N#As a corollary, the tree-width of any graph G that does not contain k vertex-disjoint cycles of length at least l \u00a0 is of order O(kl+klog\u2061k)O(kl+klog\u2061k). This is also optimal up to constant factors and ansrs another question of Birmele, Bondy, and Reed (2007)."}]