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Pancyclicity of Hamiltonian graphs

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arxiv 2209.03325 v2 pith:Y6PEVE7R submitted 2022-09-07 math.CO

Nemanja Draganić , David Munhá Correia , Benny Sudakov This is my paper · ORCID
classification math.CO
keywords graphhamiltonianpancyclicverticescontainsalreadyasymptoticallybest
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abstract

An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices, and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. In 1972, Erd\H{o}s conjectured that every Hamiltonian graph with independence number at most $k$ and at least $n = \Omega(k^2)$ vertices is pancyclic. In this paper we prove this old conjecture in a strong form by showing that if such a graph has $n = (2+o(1))k^2$ vertices, it is already pancyclic, and this bound is asymptotically best possible.

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  1. P\'osa rotation through a random permutation

    math.CO 2025-02 accept novelty 8.0 of 10

    The minimum degree threshold for a graph perturbed by a random 2-factor to become Hamiltonian is asymptotically sqrt(n log n / 2).

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