On the resilience of Hamiltonicity and optimal packing of Hamilton cycles in random graphs

TL;DR

This paper investigates the resilience of Hamiltonicity in random graphs, establishing new bounds on how much edges can be removed while preserving Hamiltonian cycles and analyzing optimal packing of such cycles.

Contribution

It introduces a generalized notion of resilience for Hamiltonicity in random graphs and improves bounds on local resilience and Hamilton cycle packing.

Findings

High probability of Hamiltonicity after edge removal under certain conditions

Improved bounds on local resilience of random graphs for Hamiltonicity

Extended range for the existence of edge-disjoint Hamilton cycles in random graphs

Abstract

Let $\bk=(k_1,...,k_n)$ be a sequence of $n$ integers. For an increasing monotone graph property $\mP$ we say that a base graph $G=([n],E)$ is \emph{$\bk$-resilient} with respect to $\mP$ if for every subgraph $H\subseteq G$ such that $d_H(i)\leq k_i$ for every $1\leq i\leq n$ the graph $G-H$ possesses $\mP$. This notion naturally extends the idea of the \emph{local resilience} of graphs recently initiated by Sudakov and Vu. In this paper we study the $\bk$-resilience of a typical graph from $\GNP$ with respect to the Hamiltonicity property where we let $p$ range over all values for which the base graph is expected to be Hamiltonian. In particular, we prove that for every $\epsilon>0$ and $p\geq\frac{\ln n+\ln\ln n +\omega(1)}{n}$ if a graph is sampled from $\GNP$ then with high probability removing from each vertex of "small" degree all incident edges but two and from any other vertex at most a $(\frac{1}{3}-\epsilon)$-fraction of the incident edges will result in a Hamiltonian graph. Considering this generalized approach to the notion of resilience allows to establish several corollaries which improve on the best known bounds of Hamiltonicity related questions. It implies that for every positive $\epsilon>0$ and large enough values of $K$, if $p>\frac{K\ln n}{n}$ then with high probability the local resilience of $\GNP$ with respect to being Hamiltonian is at least $(1-\epsilon)np/3$, improving on the previous bound for this range of $p$. Another implication is a result on optimal packing of edge disjoint Hamilton cycles in a random graph. We prove that if $p\leq\frac{1.02\ln n}{n}$ then with high probability a graph $G$ sampled from $\GNP$ contains $\lfloor\frac{\delta(G)}{2}\rfloor$ edge disjoint Hamilton cycles, extending the previous range of $p$ for which this was known to hold.