Hamiltonicity in random graphs is born resilient

TL;DR

This paper proves that in the random graph process, graphs with minimum degree at least 2 are almost surely resiliently Hamiltonian up to a certain threshold, strengthening previous results on Hamiltonicity resilience.

Contribution

It establishes the almost sure resilience thresholds for Hamiltonicity in the random graph process, extending prior work by Lee, Sudakov, and others on the resilience of Hamiltonian properties.

Findings

Graphs with minimum degree ≥ 2 are (1/2−ε)-resiliently Hamiltonian.

Graphs are not (1/2+ε)-resiliently Hamiltonian almost surely.

Every non-empty k-core with large enough k is (1/2−ε)-resiliently Hamiltonian.

Abstract

Let $\{G_M\}_{M\geq 0}$ be the random graph process, where $G_0$ is the empty graph on $n$ vertices and subsequent graphs in the sequence are obtained by adding a new edge uniformly at random. For each $\varepsilon>0$, we show that, almost surely, any graph $G_M$ with minimum degree at least 2 is not only Hamiltonian (as shown by Bollob\'as), but remains Hamiltonian despite the removal of any set of edges, as long as at most $(1/2-\varepsilon)$ of the edges incident to each vertex are removed. We say that such a graph is $(1/2-\varepsilon)$-resiliently Hamiltonian. Furthermore, for each $\epsilon>0$, we show that, almost surely, each graph $G_M$ is not $(1/2+\varepsilon)$-resiliently Hamiltonian. These results strengthen those by Lee and Sudakov on the likely resilience of Hamiltonicity in the binomial random graph. For each $k$, we denote by $G^{(k)}$ the (possibly empty) maximal subgraph with minimum degree at least $k$ of a graph $G$. That is, the $k$-core of $G$. Krivelevich, Lubetzky and Sudakov have shown that, for each $k\geq 15$, in almost every random graph process $\{G_M\}_{M\geq 0}$, every non-empty $k$-core is Hamiltonian. We show that, for each $\varepsilon>0$ and $k\geq k_0(\varepsilon)$, in almost every random graph process $\{G_M\}_{M\geq 0}$, every non-empty $k$-core is $(1/2-\varepsilon)$-resiliently Hamiltonian, but not $(1/2+\varepsilon)$-resiliently Hamiltonian.