The extremal function for cycles of length ℓ mod k [article]

Benny Sudakov, Jacques Verstraete
2016 arXiv   pre-print
Burr and Erdős conjectured that for each k,ℓ∈ Z^+ such that k Z + ℓ contains even integers, there exists c_k(ℓ) such that any graph of average degree at least c_k(ℓ) contains a cycle of length ℓ mod k. This conjecture was proved by Bollobás, and many successive improvements of upper bounds on c_k(ℓ) appear in the literature. In this short note, for 1 ≤ℓ≤ k, we show that c_k(ℓ) is proportional to the largest average degree of a C_ℓ-free graph on k vertices, which determines c_k(ℓ) up to an
more » ... te constant. In particular, using known results on Turán numbers for even cycles, we obtain c_k(ℓ) = O(ℓ k^2/ℓ) for all even ℓ, which is tight for ℓ∈{4,6,10}. Since the complete bipartite graph K_ℓ - 1,n - ℓ + 1 has no cycle of length 2ℓ mod k, it also shows c_k(ℓ) = Θ(ℓ) for ℓ = Ω( k).
arXiv:1606.08532v1 fatcat:26ryzcr6afhqznf6fj2czj6mby