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The Ramsey numbers of cycles imply that every 2‐edge‐colored complete graph on n vertices contains monochromatic cycles of all lengths between 4 and at least . We generalize this result to colors by showing that every k‐edge‐colored complete graph on vertices contains ‐edge‐colored cycles of all lengths between 3 and at least .
For an integer ℓ at least 3, we prove that if G is a graph containing no two vertex‐disjoint circuits of length at least ℓ, then there is a set X of at most vertices that intersects all circuits of length at least ℓ. Our result improves the bound due to Birmelé, Bondy, and Reed (The Erdős–Pósa property for long circuits, Combinatorica 27 (2007), 135–145) who conjecture that ℓ vertices...
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