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Research seminar in Discrete Mathematics
Hamilton cycles in pseudorandom graphs: Dirac’s theorem and approximate decompositions.
Hamilton cycles in pseudorandom graphs: Dirac’s theorem and approximate decompositions.
组织者
马杰
, 本杰明·苏达科夫
演讲者
Jaehoon Kim
时间
2025年10月21日 17:05 至 18:15
地点
Online
线上
Zoom 787 662 9899
(BIMSA)
摘要
Dirac’s classical theorem asserts that, for $n\ge 3$, any $n$-vertex graph with minimum degree at least $n/2$ is Hamiltonian. Furthermore, if we additionally assume that such graphs are regular, then, by the breakthrough work of Csaba, Kühn, Lo, Osthus and Treglown, they admit a decomposition into Hamilton cycles and at most one perfect matching, solving the well-known Nash‑Williams conjecture.
In the pseudorandom setting, it has long been conjectured that similar results hold in much sparser graphs.
We prove two overarching theorems for graphs that exclude excessively dense subgraphs, which yield asymptotically optimal resilience and Hamilton‑decomposition results in sparse pseudorandom graphs. In particular, our results imply that for every fixed $\gamma>0$, there exists a constant $C>0$ such that if $G$ is a spanning subgraph of an $(n,d,\lambda)$-graph satisfying
$\delta(G)\ge\bigl(\tfrac{1}{2}+\gamma\bigr)d$ and $d/\lambda\ge C,$ then $G$ must contain a Hamilton cycle.
Secondly, we show that for every $\varepsilon>0$, there is $C>0$ so that any $(n,d,\lambda)$-graph with $d/\lambda\ge C$ contains at least $\bigl(\tfrac{1}{2}-\varepsilon\bigr)d$ edge‑disjoint Hamilton cycles, and, finally, we prove that the entire edge set of $G$ can be covered by no more than $\bigl(\tfrac12+\varepsilon\bigr)d$ such cycles. All bounds are asymptotically optimal and significantly improve earlier results on Hamiltonian resilience, packing, and covering in sparse pseudorandom graphs. This is joint work with Nemanja Draganic, Hyunwoo Lee, David Munha Correia, Matias Pavez-Signe and Benny Sudakov.
In the pseudorandom setting, it has long been conjectured that similar results hold in much sparser graphs.
We prove two overarching theorems for graphs that exclude excessively dense subgraphs, which yield asymptotically optimal resilience and Hamilton‑decomposition results in sparse pseudorandom graphs. In particular, our results imply that for every fixed $\gamma>0$, there exists a constant $C>0$ such that if $G$ is a spanning subgraph of an $(n,d,\lambda)$-graph satisfying
$\delta(G)\ge\bigl(\tfrac{1}{2}+\gamma\bigr)d$ and $d/\lambda\ge C,$ then $G$ must contain a Hamilton cycle.
Secondly, we show that for every $\varepsilon>0$, there is $C>0$ so that any $(n,d,\lambda)$-graph with $d/\lambda\ge C$ contains at least $\bigl(\tfrac{1}{2}-\varepsilon\bigr)d$ edge‑disjoint Hamilton cycles, and, finally, we prove that the entire edge set of $G$ can be covered by no more than $\bigl(\tfrac12+\varepsilon\bigr)d$ such cycles. All bounds are asymptotically optimal and significantly improve earlier results on Hamiltonian resilience, packing, and covering in sparse pseudorandom graphs. This is joint work with Nemanja Draganic, Hyunwoo Lee, David Munha Correia, Matias Pavez-Signe and Benny Sudakov.